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Transcript
00:03:560Paolo Guiotto: Okay, so today we are going to do some exercises still on convergence.
00:09:770Paolo Guiotto: And, we also… actually, we start with this, we introduce a…
00:15:120Paolo Guiotto: Another type of convergence that will… turns out to be weaker than the previous ones, which is the so-called convergence
00:23:990Paolo Guiotto: in distribution.
00:33:90Paolo Guiotto: Or, also called the week.
00:36:640Paolo Guiotto: convergence.
00:38:960Paolo Guiotto: It's because this convergence, at least for us, is the weakest we consider.
00:46:520Paolo Guiotto: Now, the idea… Would be the following, potentially, So, we… We want… to… Give…
01:05:370Paolo Guiotto: A new definition.
01:10:200Paolo Guiotto: Off.
01:11:570Paolo Guiotto: convergence.
01:13:710Paolo Guiotto: Because it is natural to think that Xan would converge to X,
01:21:120Paolo Guiotto: If and all if, let's say, the relevant probabilities related to XN and X are convergent, so something like probability that XN belongs to E converges to the probability that X belongs to E. This is the idea.
01:41:660Paolo Guiotto: Now… This is, for every e-borel set.
01:51:280Paolo Guiotto: Now, this is, of course, is a definition we could give.
01:56:590Paolo Guiotto: But this definition would be a little bit too restrictive, because with this definition remark, with the… This… definition…
02:17:220Paolo Guiotto: even, let's say, a stupid case like this one, take Xn constantly equal to 1 over n. So, random variable constant equal to the value 1 over n. That, of course, converges
02:32:680Paolo Guiotto: in any sense, so almost surely, and I just mentioned these two.
02:38:180Paolo Guiotto: Because they are not equivalent, but they are stronger than convergence in probability.
02:43:410Paolo Guiotto: So, we'll see easily that this goes to zero. But…
02:47:510Paolo Guiotto: If we compute the probability that Xn belongs to E, we see that this does not converge to the probability that X, which is 0, belongs to E, at least not for all sets. For example, we have that BAT.
03:05:620Paolo Guiotto: We have, since X is concentrated in 0, so the probability that X belongs to E means the law of X on the set E, in this case, the law of X, if X is concentrated, is a delta Dirac. So, we can say that, for example, if the set E contains
03:26:760Paolo Guiotto: 0, so, we have that, the probabilities, the probability that, X belongs to E is, 1.
03:38:850Paolo Guiotto: If E does not contain zero, the probability that X belongs to E is, is 0.
03:47:530Paolo Guiotto: So, basically, these numbers you have here at the right are 0, 1, and so if you take, for example, this E, if… so…
03:57:510Paolo Guiotto: If, E is, for example, the interior real line without zero, Which is a Borisato.
04:07:650Paolo Guiotto: we have that, the… the probability that XN
04:13:800Paolo Guiotto: belongs to E is the probability that XN belongs to R minus
04:21:910Paolo Guiotto: 0. But Xn is constantly equal to 1 over n, so 1 over n belongs
04:28:700Paolo Guiotto: R minus 0, so that probability is 1.
04:32:550Paolo Guiotto: while the probability that X belongs to E,
04:37:00Paolo Guiotto: Since X is now constantly equal to 0, 0 is not in E. With this E, this would be 0. So these probabilities cannot converge to this one.
04:48:540Paolo Guiotto: So this definition is a reasonable definition of convergence, but
04:55:390Paolo Guiotto: in fact, it seems to be too strong, because even a stupid case like this one, where we should have convergence, it turns out that this is not convergent according to that definition. So, we have to modify, in other words, this…
05:12:850Paolo Guiotto: definition. And,
05:16:760Paolo Guiotto: Now, the idea is that this becomes true if we limit a bit the class of sets, or Boral sets E, that we consider. Now, in the previous example, in…
05:32:980Paolo Guiotto: previous… Example… We have that, replacing
05:43:830Paolo Guiotto: The, let's say, definition, start, this one.
05:48:360Paolo Guiotto: By a weaker definition, which is the following. Replacing star by…
05:55:240Paolo Guiotto: So, we pretend a little bit less than this, not for every Borel set, but for a sub…
06:03:240Paolo Guiotto: class of Borel sets. For example, this could be this one, so we say that XN converges to X if the probability that… sorry, Xn belongs to E converges to the probability that
06:21:910Paolo Guiotto: X belongs to E. This also, for every E, which is open.
06:31:180Paolo Guiotto: Open in…
06:33:590Paolo Guiotto: or if you prefer, because at the end it will be the same, equivalently, that probability that XN belongs to some open interval, AB,
06:44:370Paolo Guiotto: converges to the probability that X belongs to the same open interval AB. An open interval is a particular type of open set. An open set that, for example, you do at the union of two open intervals, you get an open set, but it is not an open interval.
07:01:620Paolo Guiotto: But the point is that since open intervals can be
07:07:930Paolo Guiotto: open sets can be written as unions of open intervals, basically, if it holds for open intervals, it holds also for open sets. Now, let's see that if we replace the previous definition with this one, then…
07:28:460Paolo Guiotto: with this definition.
07:34:820Paolo Guiotto: the XN.
07:36:440Paolo Guiotto: constantly equal to 1 over N actually goes to 0, so, which is the variable we call X. Because with this definition, we have that. Let's imagine that we computed the probability that Xn belongs to the interval AB.
07:55:870Paolo Guiotto: And we wonder if this goes to the probability that X belongs to the interval AB.
08:04:480Paolo Guiotto: It is convenient to discuss this, it is convenient to start from this number, because this number is 01, depending on the interval AB contains 0 or not, okay? So we have two cases.
08:17:280Paolo Guiotto: So, K's one… is the zero con… is in the interval AB.
08:25:730Paolo Guiotto: Now, this is different from the previous case, the previous discussion, because now, if 0 is contained in an interval AB like this one, since the interval is open, and that's its fundamental assumption here.
08:40:919Paolo Guiotto: We can say that sooner or later, also, well, 1 over n is that right, because it is positive, it's about here.
08:50:190Paolo Guiotto: Sooner or later, also 1 over n will become in… will be in that interval. So… for… And… Larger.
09:01:840Paolo Guiotto: also 1 over N belongs to the interval AB, and this is important because when we compute the probability that Xn belongs to the interval AB,
09:14:590Paolo Guiotto: Since Xn is constantly equal to 1 over n.
09:19:190Paolo Guiotto: For n large, 1 over n is in the interval, so this is a true event. This is a true-false event, because also Xn is a concentrated variable, so it has a direct delta distribution.
09:32:840Paolo Guiotto: So this one is 1, that goes to 1, which is the probability that X belongs to AB.
09:40:740Paolo Guiotto: In case 2,
09:45:880Paolo Guiotto: So if 0 is not in the interval AB,
09:50:510Paolo Guiotto: The figure is the following. We have zero. We have the interval AB somewhere, but does not contain the 0. So let's imagine that is on the…
10:01:420Paolo Guiotto: right side.
10:03:260Paolo Guiotto: The same is for the left side. So this is the interval AB. What we see here.
10:08:460Paolo Guiotto: that, of course, the probability that X belongs to AB, since X is 0,
10:16:270Paolo Guiotto: constantly, this is 0, because 0 is not in AB, so it's an impossible event. Probability is 0.
10:23:780Paolo Guiotto: But what about the probability that Xn belongs to the interval AB? Now, we know that Xn is constantly equal to 1 over N.
10:32:710Paolo Guiotto: And since 1 over n is going to 0, so we have that for n large, 1 over n is never in the interval AB, because it does, like, something like this. It moves to the left to 0.
10:45:90Paolo Guiotto: So it never… it may be for N big, it is… well, at most, it can be equal to 1, you know? Maybe 1 is here, but maybe 1 half is here. But let's say that for N large, this number, 1 over M, won't be in the interval AB.
11:02:10Paolo Guiotto: So, we can say that, for… for N large.
11:09:390Paolo Guiotto: Now, 1 over n is not in the interval AB, and therefore, since this is constantly equal to 1 over n, this probability is 0 that goes to that 1.
11:21:990Paolo Guiotto: Okay? Because for n, large means for every n greater than some initial capital N, no? So this is a sequence which is definitely constant equal to zero.
11:36:170Paolo Guiotto: Okay, so, with this, let's say,
11:43:350Paolo Guiotto: restriction of the definition, that means it's a weakening, because we do not pretend that it is true for every Borel set, but only for
11:52:730Paolo Guiotto: this part of Borel sets, the open sets, the property becomes true, okay? So now we could give the definition in this way. We say that XN goes to X in distribution, or in weak sense, if this property happens.
12:09:530Paolo Guiotto: Now, this is a possibility to give the definition. However, we could also give in this form, just with intervalsa,
12:25:60Paolo Guiotto: I don't know, mmm…
12:27:360Paolo Guiotto: Well, let's give you in this way, because there are many equivalent definitions here. The point is that with this definition.
12:36:140Paolo Guiotto: is not easy to be checked, because you should compute those probabilities. Well, maybe you can express them through the CDF, hmm? So, at the end, as you will see, this becomes a check on CDF.
12:51:720Paolo Guiotto: But in general, it's, it's a little bit difficult to start checking the convergence with these definitions, so that's why
13:01:880Paolo Guiotto: Maybe it's not the best way to give the definition. So, let's proceed to find another equivalent way, at least intuitively, then we will take that as definition, because it contains this one, okay, in some sense. So, we could notice that
13:19:90Paolo Guiotto: So, the, notice, now… the probability that XN belongs to ABE.
13:30:970Paolo Guiotto: Well, AB is the open interval, means the, if we want, so we can express this as in term of low, muon Xn of the interval AB.
13:42:710Paolo Guiotto: goes to the mu of X of interval AB, so this is an equivalent characterization. So this goes to mu X
13:54:680Paolo Guiotto: of AB is just a different way to rewrite this thing, so this goes to the probability that X belongs to the interval AB, if and only if
14:06:920Paolo Guiotto: If not, if the law of XN goes to the low of X on, when evaluated on an open interval.
14:17:680Paolo Guiotto: Now, this is, interesting because it emphasizes, something here.
14:23:360Paolo Guiotto: That, at the end, to check this property, what you need is that the laws that are always defined on, in this case, we are working with the real variables, so these are measures on the real line, arboral measures.
14:42:750Paolo Guiotto: The, the law, converged in this sense.
14:47:560Paolo Guiotto: The point is that here, in principle, we could have that the variables are even not defined on the same probability space. I could have X1 defined on a first probability space, X2 on a second probability space, X3 on a third probability space. So they are all different, defined on different probability spaces, but they lost
15:09:290Paolo Guiotto: the measures mu X, mu Xn, are defined on the same structure.
15:15:30Paolo Guiotto: So, in principle, this definition allows a meaning of convergence even for variables which are not defined on the same probabilistic structure, which is not the case for all the other definitions.
15:27:880Paolo Guiotto: So when you say, so we look here, remark.
15:36:480Paolo Guiotto: with the… This… wheat.
15:42:700Paolo Guiotto: Mate, we noticed that…
15:48:470Paolo Guiotto: In principle.
15:55:650Paolo Guiotto: D, XN… could… B.
16:02:570Paolo Guiotto: defined.
16:05:150Paolo Guiotto: on different
16:11:840Paolo Guiotto: probability spaces.
16:21:350Paolo Guiotto: Because the unique thing that counts is that mu XN on interval AB goes to mu X on interval AB, because…
16:31:360Paolo Guiotto: this.
16:34:60Paolo Guiotto: Because… Even if,
16:41:890Paolo Guiotto: DXN is a random variable, so it's measurable on some probabilities based omega n, so omega n with a family FN of events with the probability PN, and all these things are different.
16:57:900Paolo Guiotto: D.
17:00:00Paolo Guiotto: They are lows.
17:05:140Paolo Guiotto: Mu Xena.
17:08:819Paolo Guiotto: probability.
17:11:160Paolo Guiotto: measures, on the same space, R with the class of Burrell sets.
17:20:460Paolo Guiotto: Okay.
17:21:690Paolo Guiotto: So this is different.
17:25:720Paolo Guiotto: this… is different.
17:34:360Paolo Guiotto: from… the… convergences.
17:43:490Paolo Guiotto: Because, for example, you take the almost sure convergence.
17:48:40Paolo Guiotto: Almost sure conversion says you have a sequence of random variables. Let's imagine that, for a moment we do not mention the probability space, but you see immediately that
17:59:850Paolo Guiotto: These variables must be defined on the same set, because we say that XN converges almost surely to an X if
18:10:520Paolo Guiotto: what is the condition? Well, we say that the limit in n of Xn of, they are functions, so this is the pointwise convergence, basically. So Xn of omega is X of omega for almost every omega.
18:27:410Paolo Guiotto: almost every omega in the space omega. So you see that the omega where they are evaluated belongs to the same space, so it cannot be that they are defined on different structures. And similarly, when you say that XN goes, for example, in L1 to X,
18:45:440Paolo Guiotto: This is, by definition, if the expectation of the modulus Xn minus X is going to zero. But this expectation is nothing but an integral. It means integral on omega modulus Xn omega minus X omega.
19:05:110Paolo Guiotto: with respect to the probability P, which is the measure of this space, this quantity goes to zero. Again, you need that these variables are defined on the same space, and even for convergence in probability.
19:19:700Paolo Guiotto: that we introduced last time. This means that we have that the probability that Xn is away from X for an error, epsilon, this goes to zero. Or, equivalently, the probability that the distance between XN
19:36:750Paolo Guiotto: And X is less or equal, it's the same. Epsilon, this goes to 1.
19:41:560Paolo Guiotto: No? So, all these cases demand that the variables are defined on the same structures.
19:47:720Paolo Guiotto: But this, this seems to be, no longer true for the case of this type of definition. So basically, this is going to be weaker definition of convergence, also because it allows a very, very,
20:06:910Paolo Guiotto: unconstrained framework. So we could have that these variables are diff… are defined on different spaces. You may now wonder, what does it mean to have this context? Well, if you think that we are just saying that the probabilities converge to the probability of the limit.
20:24:190Paolo Guiotto: It makes sense, so it's, it's something that could be… Could make sense.
20:29:780Paolo Guiotto: Okay, so this could be a second possibility, so to say, which is basically a reformulation of the first one, to say that the sequence XN goes in weak sense to X if this happens.
20:46:260Paolo Guiotto: Now, since this is also called the distribution of the variable X, this explains why this definition is also called the convergence in distribution.
20:59:80Paolo Guiotto: However, as I said, even to work with the laws is not easy, because you are still dealing with measures, okay? It would be better if we deal with functions.
21:12:750Paolo Guiotto: So now, let's come to a further step with this definition. Now, we may imagine that
21:20:280Paolo Guiotto: we noticed that… let's go back to the initial definition, which is not a good definition. We noticed that
21:36:670Paolo Guiotto: the initial
21:40:240Paolo Guiotto: I'm doing all this to explain what it will be, the final definition we take, okay? The initial definition, the initial, let's say, wrong
21:49:970Paolo Guiotto: wrong, in the sense that it is not a good definition. So that one that says Xn goes to X if and only if, the probability that Xn belongs to E goes to the probability that X belongs to E.
22:08:810Paolo Guiotto: For every eBoreL set.
22:12:970Paolo Guiotto: could also be recasted in this way. Now, we know that a way to write the probability that Xn belongs to E is, yes, is the law of X, of the set E.
22:26:150Paolo Guiotto: But also, we can use this… write this probability in terms of expectation. So we can say this is the expectation of the event where XN belongs to A. So the indicator Xn
22:40:280Paolo Guiotto: belongs to E, or equivalently, it is the expectation of the indicator of the set E evaluated on XN.
22:49:790Paolo Guiotto: In fact, that indicator is 1 if and only if Xn belongs to Eve.
22:54:600Paolo Guiotto: So this is an indicator, let's say, usually we do not write the variable here, because it's, let's say, included in the notation expectation, but this indicator here would be an indicator of omega. So the omega for which Xn of omega belongs to E, and this is a traditional indicator, so this is a real function, no?
23:18:650Paolo Guiotto: that you evaluate on external for me.
23:21:810Paolo Guiotto: Now, you may expect that… so, if this goes to this, probability that X belongs to E, this will go to the expected value of the indicator on E of X.
23:35:710Paolo Guiotto: Now, if this formula holds, we have done several times this type of augment, we can say that if this formula holds for indicators, by linear combinations, we can make true for simple functions, and then by some limit procedure, we can make true for every function of X, provided the expectation makes sense. So, this basically
24:00:90Paolo Guiotto: could be equivalent of saying that expected value of phi of XN
24:08:800Paolo Guiotto: goes to the expected value of phi of X,
24:13:150Paolo Guiotto: Now, for what fee? For generic fee? For every fee, which is just a measurable function, this is a function in the real line.
24:22:910Paolo Guiotto: So, is, is measurable with respect to the Borel class, because,
24:29:80Paolo Guiotto: This is the reference,
24:33:370Paolo Guiotto: class for this type of functions, such that the expectations make sense, so the function is in L1.
24:42:360Paolo Guiotto: Okay, now… What should be… we have… we said that this definition is, however, not accessible because of
24:50:490Paolo Guiotto: that trivial case that makes this not interesting. We said, to make it an interesting definition, we should, we should take, restrict the class of sets E that we consider here, for example, to open sets, or even less, to intervals, that basically is equivalent.
25:12:840Paolo Guiotto: Okay?
25:13:900Paolo Guiotto: Now, what happens here? So, this property, which is the wrong definition, is equivalent to this one with every function phi, which is measurable.
25:26:340Paolo Guiotto: Now, what corresponds to this,
25:30:530Paolo Guiotto: to this property, so that the probability that XN belongs to an open interval goes to the probability that X belongs to the same open interval. Well, this means that since here we are restricting the class of sets E, the barrel sets E,
25:45:180Paolo Guiotto: In that equivalent characterization, there will be a restriction on the class of functions phi.
25:51:780Paolo Guiotto: And, so this function must be better than just a measurable function.
25:57:360Paolo Guiotto: And it turns out that, let's say, you may imagine by some kind of approximation idea. So, if you… if you look at the indicator of an open interval, AB is something like this. So, you have your open interval, AB,
26:15:140Paolo Guiotto: And the indicator is 1 here, 0 at the endpoints, and 0 down here.
26:21:820Paolo Guiotto: So, our good definition would correspond, to take as function phi an indicator of an open interval.
26:31:170Paolo Guiotto: Now, you may imagine that if this is possible, it should be possible by doing an approximation of the indicator by a good function, which is a continuous function. We can always imagine that, well, we can do linearly, for example, something like this, and the measure of the excess probably would disappear.
26:50:230Paolo Guiotto: Now, this explains, finally, the definition we use for this convergence. So, we say that… XN.
27:00:770Paolo Guiotto: goes to X in distribution.
27:04:370Paolo Guiotto: Or… in… weak.
27:08:930Paolo Guiotto: sensor.
27:12:280Paolo Guiotto: If… The, expected value of a fee of XL,
27:21:700Paolo Guiotto: Go to the expected value of feet of X, huh?
27:27:850Paolo Guiotto: Now, for which phi? For every function phi, which is continuous on the real line, and bounded.
27:39:830Paolo Guiotto: Bounded, because the indicator is bounded.
27:42:840Paolo Guiotto: And, so continuous, because if it holds for continuous function, it can be proved. This is a little bit technical, I will… I will, I will not do this proof. So, in fact, this definition is equivalent to the other one. So, proposition…
28:02:900Paolo Guiotto: This definition, so, R.
28:08:860Paolo Guiotto: equivalent.
28:11:770Paolo Guiotto: Number one, XN converges to X in distribution. Number two.
28:19:530Paolo Guiotto: the probability that XN belongs to the interval AB,
28:26:730Paolo Guiotto: converges to the probability that X belongs to the open interval AB, for every open interval AB.
28:35:910Paolo Guiotto: Content in art.
28:40:210Paolo Guiotto: So these two are equivalent, so this can be done with a little bit of hard work that it's not particularly interesting for us.
28:49:350Paolo Guiotto: An interesting fact that comes for free from this definition here is that, well, let's first check on the example.
29:01:50Paolo Guiotto: without using this equivalence, no? Let's check on the example Xn equal to 1 over N, the trivial example. Let's see if this goes in distribution to 0.
29:15:440Paolo Guiotto: Which is our X.
29:19:280Paolo Guiotto: Let's check this.
29:28:390Paolo Guiotto: using… the… definition.
29:34:540Paolo Guiotto: with this characteristic property. So we already checked that the probability that XN belongs to the open interval AB goes to the probability that X belongs to the open interval. We check a book. But now we use the definition, which is…
29:52:120Paolo Guiotto: Slightly different.
29:55:740Paolo Guiotto: Now, we have to compute the expected value of phi XN.
30:01:270Paolo Guiotto: And show that this goes to the expected value of phi of X.
30:08:440Paolo Guiotto: For every fee, continuous, and bounded. I added this B2, means continuous.
30:18:200Paolo Guiotto: And… bounded.
30:22:780Paolo Guiotto: Okay, now, this is easy, because if you look, the expected value of phi Xn is what? Well, Xn is concentrated in 1 over n.
30:34:360Paolo Guiotto: So that quantity, phi of Xn, is with probability 1 constantly equal to phi of 1 over n. So we are doing the expectation of phi of 1 over n, which is a constant.
30:47:120Paolo Guiotto: So we can carry outside of the expectation, it remains the expectation of 1, which is 1. So this is just phi of 1 over n.
30:55:420Paolo Guiotto: Now, what happens when n goes to infinity?
30:58:20Paolo Guiotto: This 1 over n goes to 0, so phi is continuous, phi is continuous, and therefore phi of 1 over n will go to phi of 0.
31:07:910Paolo Guiotto: But this phi of 0, for the same reason, is the expected value of phi
31:14:620Paolo Guiotto: Where x is constantly equal to 0.
31:21:570Paolo Guiotto: So we see that the argument works, okay?
31:25:60Paolo Guiotto: So, we check, at least, that this definition on this simple example is not in contradiction with the other one. Let's say this is exactly
31:36:730Paolo Guiotto: what we did. Well, as I said, an important factor that comes basically for free from the,
31:49:750Paolo Guiotto: Almost for free, from this characterization, is that we can characterize weak convergence, or convergence in distribution, through pointwise convergence of the characteristic functions.
32:03:890Paolo Guiotto: So in certain cases, when the characteristic functions can be computed to check with convergence, we just need to check the point-wise convergence, which is something easy, because it's just a convergence point by point. So it happens that
32:19:870Paolo Guiotto: Xn converges in distribution to X if and only if the characteristic function of XN
32:29:470Paolo Guiotto: evaluated at point C goes to the characteristic function of X at point C for every C in R.
32:38:630Paolo Guiotto: I remind that this notation, phi of xn, phi of XR the… characteristic… functions.
32:53:680Paolo Guiotto: of XN and X.
32:57:950Paolo Guiotto: Now, one implication of this is easy.
33:03:80Paolo Guiotto: The implication is this one, that if we have convergence in distribution, we have pointwise convergence of the characteristic functions.
33:13:840Paolo Guiotto: Because if you look at what is the characteristic function of XN, this is, by definition, an expectation of E to ICXN, right?
33:26:720Paolo Guiotto: So you see that this is exactly an expectation of type phi of XN, Where the function phi
33:36:390Paolo Guiotto: phi of X, if we want, is E2ICX, right?
33:42:290Paolo Guiotto: If you put XN, you get exactly that one.
33:45:150Paolo Guiotto: Now, this function is, cosine X plus i sine…
33:51:930Paolo Guiotto: CX, clearly, it is a well-defined function. On the real line, with respect to the variable X, here X is prizzed, okay? It's just a parameter. So this is a continuous function in the real line.
34:08:429Paolo Guiotto: And since… and since the modulus of phi of X
34:13:810Paolo Guiotto: is the modulus of the e to iCX. Here, both C and X are real, so that's an imaginary exponent, iCX. This is a unitary exponential, so this modulus is 1.
34:29:900Paolo Guiotto: So we can say that phi is also bounded.
34:35:449Paolo Guiotto: Okay, so phi is a continuous and bounded function, and since we have the weak convergence by definition, this says that whatever is the continuous and bounded function phi, expectation of phi Xn goes to expectation of phi of X. So, we can say that
34:55:250Paolo Guiotto: the, characteristic function phi XNC, which is the expectation of this, lowercase phi of, capital XN.
35:09:360Paolo Guiotto: This goes, by, convergence.
35:15:500Paolo Guiotto: in distribution… to the expectation of the same function, but in the variable x.
35:23:510Paolo Guiotto: And this is exactly the characteristic function of capital X.
35:29:760Paolo Guiotto: This also, whatever is C, fixed the, so for every C.
35:33:760Paolo Guiotto: Riyadh.
35:35:40Paolo Guiotto: Of course, we have also a convergence in distribution for an array, for a multivariate random variable, same definition, same properties.
35:44:910Paolo Guiotto: For the vice versa, this is a little bit more technical, because we have to show that if we have convergence in distribution, you have also… so, sorry, if you have convergence of… pointwise convergence of the characteristic function, you have convergence in distribution.
36:01:570Paolo Guiotto: Now, this is basically similar to what we have seen for the uniqueness theorem, because for the vice versa.
36:17:550Paolo Guiotto: So we should prove that the expected value of a generic function phi of Xn goes to the expectation of phi of X, no? This is what we have to do. We noticed that
36:33:520Paolo Guiotto: We… Nautilus.
36:37:320Paolo Guiotto: That,
36:38:850Paolo Guiotto: If we take a special function phi.
36:41:870Paolo Guiotto: So, if phi is the Fourier transform of an L1 function, where C is L1, and this set is the ordinary Fourier transform, okay? L1 Fourier transform of the function P,
37:00:740Paolo Guiotto: So, with this particular type of phi, now, we know that Riemann-Lebag theorem, the Fourier transform of an L1 function is continuous and bounded, so we can take this as phi, so it's a spatial type.
37:17:890Paolo Guiotto: But we know also that it's not true that every continuous and bounded function is the Fourier transform of something, because even a constant, a constant equal to 1 is continuous and bounded, and it cannot be a Fourier transform, because Fourier transform, they go to 0. But however, let's say that with this type of function, we have that the expected value of
37:38:20Paolo Guiotto: fee of this particular… of Xena.
37:42:870Paolo Guiotto: Well, this is, we, we could say it is, the, integral on, Rn, sorry, on R.
37:52:860Paolo Guiotto: Of the function phi.
37:56:260Paolo Guiotto: say, of X in… versus the law of XNN. Now, if this is a Follier transform, so the hat of a certain function C,
38:09:870Paolo Guiotto: We were reminded of this thing that was called the duality Lemma. We can move the hat on the other factor, in this case, on the measure. And what happens when we move the hat? So this is the duality.
38:25:760Paolo Guiotto: lemma.
38:29:140Paolo Guiotto: This becomes the, the hat goes on the measure, so we have a C, of, let's use now the letter X, the transform of, the Fourier transform of the measure, mu in Dixie. We have this formula.
38:46:120Paolo Guiotto: Well, now, this is the Fourier transform of the measure, and you may remind that this is nothing but the characteristic function of XN, with the unique difference that we have to flip the sign, because for the characteristic function.
39:01:810Paolo Guiotto: normally the agreement is that, as you can see here, we do… we do use the sine plus here. For Fourier transform, we have the sine minus, but the unique difference is this one. So this is minus Xi. So at the end, we have… this is the integral on R of this function P of C,
39:19:180Paolo Guiotto: phi XN minus C, dig C.
39:24:100Paolo Guiotto: everything makes sense, because this function is, by assumption, an L1 function, and this is a characteristic function which is continuous and bounded.
39:34:160Paolo Guiotto: Continuous, and bounded.
39:37:280Paolo Guiotto: So the product is still integral. Now, we pass to the limit. Here, we can do that because we have, by assumption that the characteristic functions, they converge pointwise to the characteristic function of the limit. So this is our assumption.
39:55:200Paolo Guiotto: Okay? Because we are proving the vice versa.
39:58:250Paolo Guiotto: So, by assumption, phi of XN at minus C goes to phi of X at minus C.
40:11:290Paolo Guiotto: So we want now to pass to the limit in that integral.
40:14:610Paolo Guiotto: We can do, because, so if we take C of X times phi Xn of minus x c.
40:24:560Paolo Guiotto: well, the assumption is without the minus, but this also for every C. So, in particular, when we put minus C instead of C, we have that this product goes
40:37:00Paolo Guiotto: C here is fixed, so whatever is the value of C of C, now it goes to C of C,
40:44:230Paolo Guiotto: times the characteristic function of X evaluated at minus C. These, let's say.
40:51:340Paolo Guiotto: the characteristic functions, they go for every X.
40:55:480Paolo Guiotto: Since I am multiplying by this thing, which is an L1 function, maybe it's not always defined.
41:03:390Paolo Guiotto: But it is definitely defined almost everywhere, so I can say for almost every C in R. This almost is referred here with respect to the Lebec measure, because this comes from C, and the L1 here is the ordinary L1, L1R,
41:21:780Paolo Guiotto: So respect to the Lebec measure. So this almost everywhere means with respect to Lebag measure.
41:32:180Paolo Guiotto: Okay.
41:33:450Paolo Guiotto: This is the first condition that we have in dominated convergence. We have the point-wise convergence of the argument of the integral, this one.
41:42:110Paolo Guiotto: Then we need to dominate, and this is easy, because if we take the absolute value of C of C,
41:49:90Paolo Guiotto: times T characteristic function phi Xn minus x c.
41:54:220Paolo Guiotto: Now, the characteristic function is bounded, and it is always bounded by 1. Now, remind, it is the expectation of a unitary complex number, so this is a modulus of C of C times modulus of phi X
42:10:360Paolo Guiotto: N minus C, but this quantity is bounded by 1, so we can say that this is bounded by the models of C of C, which is by assumption L1 function in the real line.
42:23:20Paolo Guiotto: So this means that dominated convergence applies, so… From this.
42:30:950Paolo Guiotto: dominated convergence.
42:34:630Paolo Guiotto: applies… And therefore, we can pass to the limit here.
42:41:600Paolo Guiotto: So let's say, just to rewrite, this was the expectation of VXN.
42:47:370Paolo Guiotto: So, expectation of phi XN, we have written this as the integral on R, where phi was, by assumption, the Fourier transform of a function C, C in L1, R.
43:05:80Paolo Guiotto: This has been written as C of C times the characteristic function of XN at minus C.
43:14:680Paolo Guiotto: Now, we passed the limit into this integral, it is here where we apply the dominated convergence.
43:23:980Paolo Guiotto: And this goes to the, integral, on R,
43:30:900Paolo Guiotto: of C. Of C times what is the limit is phi X minus C.
43:38:740Paolo Guiotto: But for the same calculation we have done at the beginning, this is the expected value of a fee evaluated now on C.
43:48:250Paolo Guiotto: So the conclusion is, Better?
43:52:300Paolo Guiotto: We have that expected value of a phi.
43:56:40Paolo Guiotto: of XN goes to the expected value.
44:01:200Paolo Guiotto: of phi of X, huh?
44:03:550Paolo Guiotto: This also, unfortunately not yet for every phi, but at least for every function phi, which is a Fourier transform of an L1 function.
44:15:910Paolo Guiotto: Now, the second part of the proof is similar to the uniqueness theorem, so I will sketch. Now…
44:25:970Paolo Guiotto: This means, in particular.
44:30:850Paolo Guiotto: So, among these functions, I could consider those functions who are, for a transform of a special particular class, which is the Schwarz class. This,
44:42:530Paolo Guiotto: means that… This means that… the… Conclusion.
44:51:330Paolo Guiotto: In conclusion, that the expected value of phi Xn goes to the expected value of phi of X holds…
45:02:50Paolo Guiotto: For every fee.
45:04:410Paolo Guiotto: which is in the Schwarz class of R, an exaggerated class with respect to continuous function, because these are C infinity, they go fast to zero, etc.
45:14:670Paolo Guiotto: And then, by an approximation argument, I have shown you how to approximate an indicator, basically. So, and then…
45:25:300Paolo Guiotto: And… Bye.
45:27:970Paolo Guiotto: Approximation.
45:34:310Paolo Guiotto: Approximations.
45:35:680Paolo Guiotto: this… expands.
45:41:610Paolo Guiotto: to… every function fee, which is
45:46:340Paolo Guiotto: Continuous and bounded in the real line.
45:50:400Paolo Guiotto: So that now we… we skip this technical part, okay? That's not the crucial point.
45:58:750Paolo Guiotto: Okay, so… What else, it's useful to know here…
46:08:940Paolo Guiotto: Well, there is another characterization which is, useful to know, that you should keep in mind, which is respect to the
46:21:310Paolo Guiotto: CDF.
46:23:290Paolo Guiotto: Okay? Because sometimes it could be easy to compute the characteristic function, but that's a Fourier transform, so it's not always easy to do, but sometimes it's useful.
46:34:190Paolo Guiotto: it's a convenient way. Other times, you have the CDF, so these are the ingredients. So if you have a… let's say, if you would know the variables, you could proceed by computing expectations, and this has the difficulty that you have to show this for every function fee, which is continuous, so it's a generic check.
46:55:680Paolo Guiotto: That's… maybe it's not easy. This one…
46:59:990Paolo Guiotto: is a more direct check, because it does not involve any generic function, you just compute the characteristic function of DXN, and you do the pointwise limit. So, as a check, it is very easy, no? The only problem is that we must be able to compute the characteristic functions, and then to do the limit.
47:17:580Paolo Guiotto: But these are technical issues, let's say. Sometimes you may have, instead of having the characteristic function, you may have the CDF. So, what is the condition? Fortunately, there is a nice check, which is the following, R equivalent.
47:37:280Paolo Guiotto: Number one.
47:38:610Paolo Guiotto: that XN converges to X in distribution. And number two, the CDF of XN, also here we have a pointwise convergence, so it's good. So this converges pointwise to the CDF.
47:55:980Paolo Guiotto: of X, but be careful, because this is not for… necessarily for every X. This must be true for every…
48:21:70Paolo Guiotto: X, so…
48:27:530Paolo Guiotto: Okay, I will not prove this, there is a part of the proof, because also this one is quite technical, it's… it's nice, but…
48:39:440Paolo Guiotto: It's a little bit long, so, we, we just, we just skip. So, we may say that, so now let's put all the three characterizations. The, the, the, the number 3 is the previous one, so phi, X, and…
48:55:210Paolo Guiotto: at point C goes to phi X at point C, for every C, real.
49:03:670Paolo Guiotto: Well, we may say, let's add also this one, that the probability that XN belongs to the interval AB
49:14:850Paolo Guiotto: goes to the probability where X belongs to the open interval AB.
49:20:740Paolo Guiotto: for every interval AB.
49:25:890Paolo Guiotto: Well, there are… let's say there is a list of something like 7, 8 equivalent characterizations of this, okay? And, of course, we have also the definition, so this is here, expected value of phi.
49:42:210Paolo Guiotto: of XN goes to the expected value of ph.
49:49:20Paolo Guiotto: for every P, continuous and bounded.
49:54:600Paolo Guiotto: So, you may use one or the other, depending on what is the convenience, and sometimes you need also, especially about the number 4, to use this as a conclusion. So, if you have weak convergence, then you have convergence of the probabilities. This is what, for example.
50:14:110Paolo Guiotto: Happens with the case of the so-called central limit here that we will see soon.
50:20:900Paolo Guiotto: Mmm…
50:22:830Paolo Guiotto: Now, before we… yeah, before we enter into exercises, let's establish a connection between this convergence and the previous one, and…
50:34:320Paolo Guiotto: So the, convergence in distribution, or weak convergence, convergence in distribution.
50:45:10Paolo Guiotto: Or weak.
50:48:960Paolo Guiotto: convergence.
50:50:260Paolo Guiotto: Now, the name weak convergence comes actually from functional analysis.
50:55:540Paolo Guiotto: I don't know if you… it may be that you have done something with the PDEs, huh?
51:02:690Paolo Guiotto: I've never heard of this.
51:05:00Paolo Guiotto: with convergence.
51:08:40Paolo Guiotto: Okay, in any case, for your information, this is not the weak convergence, the name is used in functional analysis. The weak convergence is not this one, this would be what is called in functional analysis.
51:24:510Paolo Guiotto: Analysis… This is, the… Not the week, but the so-called week… star.
51:36:890Paolo Guiotto: convergence, which is a weaker form of convergence. It's the weakest form of a convergence, let's say.
51:44:440Paolo Guiotto: But why they call the weak convergence? Because, you know, when something is translated from analysis to probability, it takes something, but not everything. So the Fourier transform is not the Fourier transform, they change a sign. It becomes the characteristic function.
51:59:790Paolo Guiotto: Sometimes in analysis books, the characteristic function is synonymous of indicator function.
52:06:50Paolo Guiotto: at least in all textbooks, in analysis, the characteristic function means the indicator function. So, you see, there are names that, moving from one discipline to another, they just change a little bit of meaning. However.
52:23:170Paolo Guiotto: For us, we're content of the fact that… why it is called a weak convergence, because it's the weakest of the convergence we know. So, is weaker…
52:36:280Paolo Guiotto: Now, it's sufficient to connect this to convergence in probability, because almost pure convergence implies convergence in probability. L1 convergence implies convergence in probability, so we know that these two convergence, which are not equivalent, they are stronger. Now, this is what…
52:55:920Paolo Guiotto: The, the figure we have seen somewhere here last time.
53:00:90Paolo Guiotto: No? We know that L1 implies convergence in probability, almost sure convergence implies convergence in probability, and now we are at the last step here. Convergence in probability implies weak convergence, okay? So…
53:16:670Paolo Guiotto: is weaker than… convergence.
53:22:650Paolo Guiotto: in… probability.
53:26:450Paolo Guiotto: So… mmm… We can prove this proposition.
53:36:00Paolo Guiotto: if XN.
53:38:40Paolo Guiotto: goes to, X in probability.
53:42:310Paolo Guiotto: Ben… Xn goes to… X also in distribution, but of course, the vice versa is false.
53:53:270Paolo Guiotto: So they are not equivalent.
53:59:950Paolo Guiotto: Since it's already 9.35, for a moment, I want to do the proof. It's not a long proof, but it's a bit theoretical. I don't know if…
54:12:300Paolo Guiotto: We could get any idea from this,
54:16:400Paolo Guiotto: I would prefer to do some exercise now.
54:20:380Paolo Guiotto: So that you can work this along the weekend.
54:24:730Paolo Guiotto: So, okay, now let's move to some, problems.
54:32:720Paolo Guiotto: The first problem I want to do is the exercise, 855.
54:41:360Paolo Guiotto: This is actually an exercise on a convergence improbability, but then I will give you some evaluations of these.
54:48:340Paolo Guiotto: I'll prepare here.
54:50:860Paolo Guiotto: That makes convergence probability and pro-convergence in distribution.
54:55:520Paolo Guiotto: So the 855 says something of this type that looks like a rule of calculus. Now, you know that for ordinary sequences and limits, we have things like the limit of the sum is the sum of the limits, the limit of the product is the product of the limits, and so on.
55:12:320Paolo Guiotto: But with these definitions, you have to be a little bit careful, okay? Unless it is the point-wise convergence, or the L1, where the algebraic limits rule works. For convergence in probability, and mostly in convergence in distribution, it can be different. So this says that if Xn goes to X in probability.
55:33:120Paolo Guiotto: And the YN goes to Y in probability.
55:38:00Paolo Guiotto: Then, they sum XN+.
55:41:270Paolo Guiotto: YN goes to X plus Y in probability. This is basically an exercise on definitions, okay?
55:52:590Paolo Guiotto: So, what do we have to check? So we… Ava… Check.
56:01:650Paolo Guiotto: remind that you have always these two possibilities for the convergence in probability, which are equivalent. It depends, which one to use, it depends on what is the idea you have to assess something, because in both cases, you have to assess a probability.
56:17:510Paolo Guiotto: Here, we have to assess the probability that the distance between XN plus YN and X plus Y
56:25:990Paolo Guiotto: So if you want to use the straight definition, greater than epsilon, this goes to zero when n goes to infinity, and this for every epsilon positive. So epsilon is fixed, you fix an epsilon positive, and you want to show that the probability… that the gap between the two
56:42:230Paolo Guiotto: is greater than epsilon goes to zero. Or equivalently, there is an equivalent characterization if you inverse this inequality, so modulus of XN,
56:52:470Paolo Guiotto: plus Yen.
56:54:790Paolo Guiotto: minus X plus Y.
56:57:900Paolo Guiotto: less or equal than epsilon. But in this case, this means the probability that you are close to the limit, this will go to 1, okay?
57:08:570Paolo Guiotto: still, sorry, to 1, still, when n goes to plus infinity, and for every epsilon positive.
57:17:320Paolo Guiotto: So they are equivalent, it depends on what you think is better to do. If to assess, because in any case, you will have to assess. Generally, it's impossible to compute exactly things, but so in the first case, I have to show that some probability is small, in the other case, I have to show that some probability is big, basically, because 1 is the maximum possible.
57:36:790Paolo Guiotto: So it depends, because in the first case, I have to proceed by upper bounds, no? I will say probability is less than something, which is less than something, that goes to zero at the end, no?
57:47:960Paolo Guiotto: In the second case, I will do lower bounds, because I have to prove I am always below 1, no? With the probability. So I have to show that I am close to 1, means I am
57:58:380Paolo Guiotto: bigger than something which is close to 1, so I will proceed by saying probability is greater than this, than greater than that, which is greater than 1 minus something that goes to 0.
58:10:60Paolo Guiotto: This makes a difference, because when you have to assess by upper bounds.
58:15:780Paolo Guiotto: These are events, so you want to assess the probability of an event by another event, maybe, which is easier to treat.
58:23:770Paolo Guiotto: So the second event must be bigger. So you have to think by inclusion that enlarge the set, your event. You start from this event, and you say, maybe this is difficult to be assessed, but they find something larger where the probability is more, where it's easier to say that the probability is more. And here, it's the contrary, because you have to take something which is smaller.
58:48:550Paolo Guiotto: where the probability is big, okay? So you have to be… this is basically the problem with these probabilities. Of course, you may say, we could compute the probability directly, but you see, if you have to compute directly a probability like this.
59:05:930Paolo Guiotto: This is something that depends on four different variables. Xn, YN, X, and Y. So if you want to compute this probability, you need the joint distribution of these four things, that could be… for example, here is not even known.
59:21:790Paolo Guiotto: You see? So it could be complicated.
59:25:190Paolo Guiotto: And definitely, this is not the way to assess these probabilities. So now, let's think with the first one. So, of course, the first thing we could do
59:35:10Paolo Guiotto: So, I want to take this gap, let's say, greater or equal than epsilon. Now, I will rearrange the quantity inside, and I have XN minus X, because I know something about these gaps, no? Xn minus X and Yn minus Y.
59:53:730Paolo Guiotto: And I want to show that this is greater or equal than epsilon.
59:57:840Paolo Guiotto: Now, clearly, this, of course, there are the same ideas we use each time we have to prove things like this. Now, the modules of the sum is less than sum of the models, so let's see what happens here. So let's rewrite the inequality in this way. This is less than Xn minus X.
00:17:280Paolo Guiotto: plus YN.
00:19:260Paolo Guiotto: minus Y.
00:21:470Paolo Guiotto: Let's try to see what happens if we split this into the modulus of Xn minus X plus the modulus of YN minus y. This is the triangular inequality.
00:33:100Paolo Guiotto: Now, you see here, you see here,
00:38:990Paolo Guiotto: Well, no, you don't see here.
00:40:930Paolo Guiotto: Well, you see here what? That, for example, this inequality says that
00:45:900Paolo Guiotto: If you are in the event where this thing, you know, this big mass is greater than epsilon, then you are also in the event where that sum is larger than epsilon. So it means that sum larger than epsilon is bigger or smaller.
01:03:470Paolo Guiotto: Then the odd event.
01:14:570Paolo Guiotto: So, it says, this is what it's saying, if you verify this, the big modules of the… these models here, of the difference between the sums, is larger than epsilon. Then, also, this one will be larger than epsilon.
01:31:650Paolo Guiotto: So, you see, if you verify this, then you verify that. So it means that this is contained, even that is greater than epsilon. So that event is larger, and that's not good, because we have to proceed on the contrary, right?
01:48:990Paolo Guiotto: We have to proceed, bye.
01:51:110Paolo Guiotto: No, yeah, that's why. I was doing a mess. So, we have to proceed by inclusion. So, it says that the event where modulus of XN plus YN minus…
02:07:00Paolo Guiotto: X plus Y is larger than epsilon. That one for which we have to assess the probability, is…
02:17:690Paolo Guiotto: contained in the event where modules XN minus X plus
02:27:540Paolo Guiotto: modulus y n minus Y is larger than epsilon, right?
02:32:820Paolo Guiotto: Now, this is not yet completely interesting, because let's think through what we know,
02:41:620Paolo Guiotto: So these are the assumptions. So, remind that, we know We know what?
02:51:570Paolo Guiotto: Dr. Xena, goes to X in probability, so this means that the probability that Xn is away from X
03:05:430Paolo Guiotto: goes to zero, this thing goes to zero, and the same for YN. YN goes in probability to Y, so the probability that YN is
03:17:150Paolo Guiotto: away from Y.
03:20:00Paolo Guiotto: Also, this one goes to zero.
03:22:210Paolo Guiotto: And here I have the event where this sum is greater than epsilon.
03:27:410Paolo Guiotto: Now, in some way, I want to split into these two sub-events, you see?
03:34:340Paolo Guiotto: So I want to transform this into… reduce this to this to sub events, XN minus X, larger than… I do not specify for the moment what is that, YN minus Y larger than.
03:49:200Paolo Guiotto: Because we see that it's not epsilon.
03:52:680Paolo Guiotto: Of course, if I put epsilon cannot be… well, what could be the relation? If I am… if I put an epsilon here, an epsilon here, well, each of them is contained in the other, so I'm not enlarging, I am reducing.
04:07:550Paolo Guiotto: But I want to enlarge, so I want that this be contained somehow in these sets.
04:14:290Paolo Guiotto: Well, what is the remark? Here, it's a numerical remark. If you have two numbers, so remark.
04:22:540Paolo Guiotto: If you have two numbers, A plus B, they are positive, because A and B are the two moduluses, okay? A and B.
04:31:760Paolo Guiotto: They are positive, and you know that their sum is greater than absolute, then what can you say about A and B?
04:38:40Paolo Guiotto: They must be larger than the
04:42:730Paolo Guiotto: Or at least one of them must be larger.
04:54:860Paolo Guiotto: They cannot be too small, no? Because suppose that they are both zero, the sum will never be equal to epsilon, okay? Or greater.
05:02:260Paolo Guiotto: So they cannot be very small, because if they are too small, some will never be larger than epsilon. How much smaller?
05:13:760Paolo Guiotto: They cannot be both smaller than
05:17:820Paolo Guiotto: something that, when you sum, two times gives epsilon, so it is epsilon over 2. In fact, one of them must be larger than epsilon over 2.
05:29:320Paolo Guiotto: Because if… Both A Is strictly less than epsilon over 2.
05:42:710Paolo Guiotto: B is less than epsilon over 2, then you would have that A plus B would be less than epsilon.
05:50:860Paolo Guiotto: So this would be contradicting. So the trick is just here. So we can say that since one of them must be larger than epsilon over 2, it means that, as a set, the set where the sum of the two, modulus Xn minus X,
06:07:210Paolo Guiotto: plus modulus YN minus Y.
06:11:220Paolo Guiotto: is larger than epsilon. This must be contained, you see, we have to enlarge, because we are assessing from above.
06:20:540Paolo Guiotto: must be contained into the union of distance between XN and X larger than epsilon F, union with distance between YN and Y larger again than epsilon F.
06:36:480Paolo Guiotto: You see?
06:38:440Paolo Guiotto: So, now we have the key to solve the problem, because… so we started from this, and we had to assess the probability that this… the probability of this event, so we can say that.
06:53:160Paolo Guiotto: The probability that models of the sum Xn plus YM
06:59:520Paolo Guiotto: minus X plus Y is larger than epsilon.
07:05:770Paolo Guiotto: Since this is contented into this, which is contented into the other two, this will be less or equal than the probability of the union of the two, so modulus Xn minus X greater than epsilon half.
07:21:180Paolo Guiotto: union.
07:22:790Paolo Guiotto: modus YN minus Y greater than epsilon half.
07:31:200Paolo Guiotto: And this is less or equal by sub additivity, so we are still…
07:36:620Paolo Guiotto: Increasing the quantity, because we are…
07:39:940Paolo Guiotto: looking for an upper bound. It's bounded by probability that XN minus X
07:47:420Paolo Guiotto: is away from zero, so the distance is larger than epsilon, plus the probability that epsilon half, actually, modus yn minus Y
07:58:650Paolo Guiotto: Is greater or equal than epsilon half.
08:01:860Paolo Guiotto: Now we should be at the conclusion, because this by assumption goes to zero, this by assumption goes to zero, because of the convergence in probability. So they both go to zero, and therefore this quantity, which is between 0 being a probability.
08:18:359Paolo Guiotto: And this number that goes to zero.
08:21:760Paolo Guiotto: By the… the… what is the name?
08:28:479Paolo Guiotto: That's abolishment here, right? That's another name,
08:34:180Paolo Guiotto: However, this shows that the probability of
08:38:750Paolo Guiotto: This event here, greater than epsilon, goes to zero, and this is the conclusion.
08:46:109Paolo Guiotto: So… Exercise for you.
08:52:130Paolo Guiotto: What if, XN goes to X.
08:59:819Paolo Guiotto: YN goes to Y, still in probability.
09:04:830Paolo Guiotto: Is it true that, XN… YN… goes to XY in probability.
09:14:90Paolo Guiotto: So it is true that the limit of the product is the product of the limit, so think about.
09:20:390Paolo Guiotto: Another,
09:23:109Paolo Guiotto: exercise for you is, you know that it's a weakened version of this one. You know now that XN goes to X in probability.
09:34:740Paolo Guiotto: And YN goes to Y, but in a weaker sense in distribution. What about XNYN?
09:45:210Paolo Guiotto: Can we say that it goes to XY in distribution.
09:52:550Paolo Guiotto: Can we say that XNYN goes to XY improbability.
10:01:180Paolo Guiotto: So, think about this.
10:06:170Paolo Guiotto: Okay.
10:09:800Paolo Guiotto: Now… We have a nice exercise, 8, 5, 6, because it…
10:16:420Paolo Guiotto: Or maybe I will do first the 85 ATM.
10:21:300Paolo Guiotto: Which is a…
10:30:10Paolo Guiotto: Okay, this says we have a sequence XN of… it says IID. I don't know if we introduced this acronym. It is independent
10:43:240Paolo Guiotto: independent.
10:47:250Paolo Guiotto: identically.
10:52:710Paolo Guiotto: distributed.
10:57:810Paolo Guiotto: So, they are independent, and they have also the same distribution, the same CDF, the same load, the same
11:05:10Paolo Guiotto: everything.
11:06:590Paolo Guiotto: And, it says, with this, it gives you this information. The probability that XN is greater than X gives this.
11:17:710Paolo Guiotto: which is not the CDF, strictly speaking. It is 1 over root of X for every X greater or equal than 1.
11:29:230Paolo Guiotto: Then, it defines this variable, mn be the maximum of these variables, X1, to XN.
11:39:290Paolo Guiotto: So, question one is, determine the CDF of MN.
11:47:710Paolo Guiotto: Question 2.
11:50:270Paolo Guiotto: discuss… convergence… in distribution, of the sequence MN.
12:05:400Paolo Guiotto: Identifying the limit, if any.
12:09:250Paolo Guiotto: what… about… all the… convergences. This is a question I'm adding now.
12:23:540Paolo Guiotto: So the… Convergence in probability, Almost sure, etc.
12:30:630Paolo Guiotto: So, question one.
12:33:300Paolo Guiotto: Well, if you want, we can start from this amen. So, the CDF,
12:41:440Paolo Guiotto: of MN is this function, let's say, capital F, MN,
12:48:330Paolo Guiotto: of, say, X is the probability that MN
12:53:790Paolo Guiotto: is less or equal than X. No? This is by definition.
13:00:490Paolo Guiotto: Now, this is interesting because minimum, maximum are always important things in applications, so we have to compute the probability that the maximum here of X1, XN be less or equal than this value X.
13:19:100Paolo Guiotto: This is standard. Now, when the maximum is less or equal than X, the maximum is less or equal than X if and only if all these quantities are less or equal than X, because the maximum is the bigger, no? So, if and not if X1 is less or equal than X,
13:38:110Paolo Guiotto: X2 is less or equal than X, X3 is less or equal than X, XN is less or equal. Now, these commas means end, and, and, okay? Because if only one is greater than X, the maximum would be greater than X.
13:54:50Paolo Guiotto: So, we have actually, to compute the probability of this event, X1 less or equal X, X2 less or equal X, and so on.
14:03:40Paolo Guiotto: And this is an intersection, XN less would equal X.
14:08:690Paolo Guiotto: We know also that the variables are independent, right? So this means that these events are independent events, so these are independent
14:21:610Paolo Guiotto: independent.
14:23:550Paolo Guiotto: events.
14:25:880Paolo Guiotto: And therefore, the probability splits into the product, so probability that X1 is less or equal than X, probability that X2 is less or equal than X, and so on.
14:38:380Paolo Guiotto: And these are, what? These are the CDF of DXN.
14:42:950Paolo Guiotto: Since they are, they are identically distributed now, means they have the same distribution, they have the same CDF. So, in fact, this is the same function, so FX1, X, FX2, X,
15:00:380Paolo Guiotto: And so on, F, X, and F, but all these are the same thing, the same quantity.
15:05:380Paolo Guiotto: They are all equal, because identically… identic distribution.
15:17:590Paolo Guiotto: Okay? So, let's compute that distribution, and this will be the product of the same quantity and time, so the power of that.
15:25:850Paolo Guiotto: So now, what is, F, say, FX1 of X?
15:30:710Paolo Guiotto: It is the probability that X1 is less or equal than X. The problem gives the probability that X1 is larger than X, and it gives for every X greater or equal than 1.
15:45:80Paolo Guiotto: So, now, of course, we try to connect this with that. That's easy because you are less or equal than X if you are not greater than X. So you could say that this is omega minus the event X1 larger than X, and since this is a subset, we can use the difference for probability we do not have two.
16:08:640Paolo Guiotto: care too much about this, so it is probability of omega 1
16:12:170Paolo Guiotto: minus the probability where X1 is larger than X.
16:17:580Paolo Guiotto: Now, this says that this quantity is 1 over root of X when x is greater or equal than 1, so definitely we're right. It is equal to 1 minus 1 over the root of X for X greater or equal than 1.
16:32:100Paolo Guiotto: And what about the 4x less or equal than 1?
16:35:510Paolo Guiotto: Now, you may notice here, so for X less than 1,
16:40:280Paolo Guiotto: You may notice that from this formula, in particular, it follows that the value at 1
16:47:710Paolo Guiotto: Is equal to zero.
16:50:330Paolo Guiotto: And, so this is important, because if you, if you plot this function, so this function starts at 1 as 1 minus 1 over the root, so the root is increasing, 1 over the root is decreasing, minus is increasing, so it will be something like this.
17:13:300Paolo Guiotto: Let's say that this is… well…
17:15:430Paolo Guiotto: I put it on the other seat.
17:20:620Paolo Guiotto: This is one.
17:22:550Paolo Guiotto: So this is the plot of the CDF for X larger than 1. What do we know about the CDF? We know that the CDF is an increasing function, the values are always between 0 and 1, and since at 1 it is equal to 0, it means that here it must be equal to 0.
17:38:830Paolo Guiotto: Okay, so this implies that FX1 of X is equal to 0 for X less than 1.
17:48:390Paolo Guiotto: So we can say that the FX1 of X is basically 1.
17:55:100Paolo Guiotto: minus 1 over the root of X.
17:58:890Paolo Guiotto: Times, if we want the indicator of 1 plus infinity.
18:02:950Paolo Guiotto: Well, it is clear that if I take X negative, for example, that thing is not defined, but who cares? We may interpret everything times 0 is 0.
18:12:690Paolo Guiotto: Okay, so… Or, if you want to be more precise.
18:18:920Paolo Guiotto: So when we have to go back to the CDF of MN,
18:24:670Paolo Guiotto: Which is the product, so it is the same function repeated n times, so it is the power of that function.
18:33:740Paolo Guiotto: We can say that this, when X is less than 1, this is 0, because you are doing the products of zeros. When x is greater or equal to 1, you are doing the product of these quantities, so 1
18:46:540Paolo Guiotto: Minus 1 over root of X to power n.
18:52:680Paolo Guiotto: And so we computed the DCDF of this semi.
18:56:610Paolo Guiotto: About question two, now we have to, discuss,
19:01:820Paolo Guiotto: Convergence, convergence, in… distribution.
19:10:630Paolo Guiotto: of this MN.
19:13:990Paolo Guiotto: Well, since we have the CDF here, it would be reasonable to work with this characterization of the convergence in distribution, no? This property here.
19:30:460Paolo Guiotto: Okay.
19:31:850Paolo Guiotto: We could also compute the characteristic function, but this would take a little bit,
19:39:860Paolo Guiotto: A little bit more… a bit longer time, because we have to…
19:43:560Paolo Guiotto: how do we compute now the CDF, sorry, the characteristic function, if you want?
19:48:760Paolo Guiotto: We have the CDF, so we have the density by doing the derivative, and then we have to do the Fourier transform of the density, so this would be the way. And maybe we do not calculate this, because when we do the derivative, we have a function. By the way, if you see here.
20:08:290Paolo Guiotto: This is not easy, because if you look at the derivative of this function, this is N times that parenthesis to minus 1, n minus 1, which is a quantity bounded by 0 and 1, so it's irrelevant for the integrability, but then
20:23:560Paolo Guiotto: you have minus 1 over root of X, so X to minus 1 half, that is minus 1 half x2 minus 1… minus 3 half for x greater than 1. So, at the end, we have to do the written form of something like 1 minus 1 over root of X to some integer times 1 over x to…
20:43:550Paolo Guiotto: 3 halves, 4X greater than 1, so it's something that probably we don't compute, okay? So, forget of that. We here see that, clearly, that's easy, that, so, since the point is to do the point-wise limit, the only care is that
21:01:660Paolo Guiotto: you see here, we do not yet have a DX.
21:05:720Paolo Guiotto: Okay?
21:07:110Paolo Guiotto: So, the limit variable. We don't know even if the limit exists, so we don't know if the candidate limit, the X. So, this is important because the point-wise limit is
21:19:590Paolo Guiotto: for the X for which the limit CDF is continuous, so we do not have that, so how can we handle this? However, we can easily compute these limits, and this we can do for every X, fortunately.
21:32:10Paolo Guiotto: So independently of what is the candidate, so we can say that a limit when n goes to infinity of the CDF of Mn of X. Well, if X is less than 1, we are doing limits of 0, so we will get 0.
21:49:240Paolo Guiotto: So we say it is 0 for X less than 1. Otherwise, for X greater or equal than 1, we are doing the limit when n goes to plus infinity of 1 minus 1 over the root of X to the power n.
22:04:740Paolo Guiotto: Now, what about this? Remind that this is a limit for N,
22:09:410Paolo Guiotto: And it's the variable, and the X is fixed, it's a parameter. So when you fix X, you know that X greater than 1 means this number is between 0 and 1.
22:20:740Paolo Guiotto: No? Because X is greater than 1, 1 over root of X will be less than 1. You are subtracting 1 over root of X to 1, so this number will be between 0 and 1.
22:30:940Paolo Guiotto: And it cannot be 1, because to be 1, you should subtract 0, no? So this is actually a number between 0 and strictly less than 1. This is important, because when you do this to power n, and you send n to infinity, you get 0.
22:47:900Paolo Guiotto: So, the limit is also 0 for X greater or equal than 1.
22:53:200Paolo Guiotto: So, it means that the point-wise limit is 0 for every X.
22:59:100Paolo Guiotto: So at least the point-wise limit exists.
23:02:520Paolo Guiotto: And now, this point-wise limit should be… so, if XN… sorry, MN
23:10:680Paolo Guiotto: If MN converge in distribution to M, then we should have that the point-wise limit of the CDF of MN
23:23:920Paolo Guiotto: would converge to the value of the CDF of this Unknown M at point X,
23:32:20Paolo Guiotto: for every X continuity point of this unknown CDF.
23:40:620Paolo Guiotto: But we do know that this goes to zero for every X.
23:45:840Paolo Guiotto: So, we would get that… This function, FM of x, should be equal to 0,
23:53:440Paolo Guiotto: not necessarily for every X, but for every X, which is a continuity point, for every X, continuity point.
24:02:740Paolo Guiotto: of… Fm.
24:06:540Paolo Guiotto: Now… Does this mean that FM should be equal to zero?
24:12:280Paolo Guiotto: Because it is clear that if FM is equal to 0, there cannot be any random variable with that CDF, you see? Because CDF at infinity must be 1.
24:23:500Paolo Guiotto: So, this would mean that there cannot be M, there cannot be limit.
24:28:70Paolo Guiotto: So it means that the sequence is not convergent. Now, the point is, can we say that this is the, let's say, the delicate step, then FM…
24:41:430Paolo Guiotto: Well, we just, maybe it's not… maybe, FM is not identical in zero, but is there anything that can be said about this which is, which is impossible now? Because, you see,
24:56:640Paolo Guiotto: Now…
24:58:170Paolo Guiotto: It is clear if I knew that FM is continuous, this would be for every X, so this would be a contradiction. But FM could not be continuous, so what can be said about this? Another point is, what are the discontinuity points of SCDF in general?
25:18:750Paolo Guiotto: I left a problem that was in chapter 3, I think, a problem that shows… Its number is…
25:29:870Paolo Guiotto: 3, 4… something.
25:39:280Paolo Guiotto: Which is something interesting by its own to know.
25:43:850Paolo Guiotto: Yes, the exercise… Is 3, 4, 9, huh?
25:50:580Paolo Guiotto: Annie.
25:52:360Paolo Guiotto: CDF.
25:55:20Paolo Guiotto: Oz.
25:56:610Paolo Guiotto: At most,
26:00:660Paolo Guiotto: accountable.
26:05:330Paolo Guiotto: Set.
26:08:900Paolo Guiotto: discontinuities.
26:16:900Paolo Guiotto: So, there cannot be too many discontinuities. At most, it's a countable set. Now, why this yields to a contradiction here? Because…
26:28:490Paolo Guiotto: Suppose that, FM…
26:32:510Paolo Guiotto: at some point X0. So the question is, is this implying that this FM is identically zero? Now, suppose that FM at some point X0 is not 0. Since it is between 0, 1, means it is positive at some point X0.
26:49:150Paolo Guiotto: Now, what would you deduce if this happens? So I'm saying, suppose that whatever is this CDF, there is a point, X0, where the value of the CDF is strictly positive here. This is FM of X0. But then, you know, the CDF is an increasing function.
27:09:330Paolo Guiotto: Whatever it happens after X0, the value of F will be still positive, because it is increasing. Since DFM is increasing, I would have that FM
27:23:280Paolo Guiotto: of X will be greater than FM of X0, this for every X greater or equal than X0.
27:32:400Paolo Guiotto: So this means in particular, this is strictly positive.
27:36:200Paolo Guiotto: So, if the FM is different from zero at one single point, it is definitely, you know, different from zero for all points greater than this one.
27:47:150Paolo Guiotto: And, so this means that, so, look at this now. It says that FMX is going to zero for every X.
27:56:950Paolo Guiotto: So if this is not the value of FM, so if FM is non-zero.
28:02:810Paolo Guiotto: It means that all points where this is different from zero are these continuity points. Otherwise, if they were continuity points, this would be zero. So this, in particular means that since FMN
28:19:550Paolo Guiotto: at this point, X. On one side is going to 0, and on the other side, FM of X is different from 0, so I can say that this will not go to this one. So it means that for every X larger than X0,
28:37:550Paolo Guiotto: Is a discontinuity point.
28:42:380Paolo Guiotto: of FM. Otherwise.
28:45:940Paolo Guiotto: we would have that FMX would go to F… FMNX would go to FMX at those points. But then… then this means that this set, X0 to plus infinity.
29:02:60Paolo Guiotto: all… discontinuities.
29:05:790Paolo Guiotto: discontinuity point.
29:08:110Paolo Guiotto: for FM, and this is impossible with this fact that the CDF has, at most, a comparable set of discontinuities.
29:19:90Paolo Guiotto: Which is… Which… contradicts.
29:27:370Paolo Guiotto: D.
29:30:580Paolo Guiotto: bets.
29:33:120Paolo Guiotto: M. Oz.
29:35:720Paolo Guiotto: at Nosta.
29:39:510Paolo Guiotto: accountable.
29:42:180Paolo Guiotto: Set.
29:44:180Paolo Guiotto: of these continuities.
29:48:90Paolo Guiotto: So this implies at the end that MN cannot converge in distribution to anything.
30:00:500Paolo Guiotto: Okay.
30:01:690Paolo Guiotto: this morning, we started 840.
30:07:460Paolo Guiotto: Right? Okay, so we are in time now.
30:11:420Paolo Guiotto: So, for the next time…
30:15:490Paolo Guiotto: I warmly suggest you to do.
30:19:690Paolo Guiotto: do exercises 8, 5, Sixer, this is very important.
30:27:810Paolo Guiotto: not because of the exercise, but it's… it's a standard, let's say, I think.
30:35:900Paolo Guiotto: Well, maybe the third part could be… because the almost sure convergence is always A little bit complicated, so…
30:44:440Paolo Guiotto: And the other one is 857…
30:52:660Paolo Guiotto: Okay, do this too. And the exercise, also, I left you yesterday, here in the end.
31:02:970Paolo Guiotto: Which is the same exercise we just did. It changed the distribution of DXN, so the CDF you have already done, but here you have to discuss the convergence of MN divided by N, okay?
31:17:310Paolo Guiotto: So, you have to pass through the same,
31:24:180Paolo Guiotto: way, and then to discuss the convergence, or mutual convergence. This is a little bit tricky, this one.
31:31:220Paolo Guiotto: So, however, try to do the first part, at least the first two questions. The first one we have already done.
31:37:790Paolo Guiotto: Okay.
31:39:00Paolo Guiotto: Let me stop here.