Talk:Convergence of random variables

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Proposed deletion of Lévy's convergence theorem[edit]

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The article Lévy's convergence theorem has been proposed for deletion because of the following concern:

This is basically a duplicate of a part of Dominated convergence theorem, but with a wrong name: no reference has been found where Lévy is credited. See the talk page for more info.

While all contributions to Wikipedia are appreciated, content or articles may be deleted for any of several reasons.

You may prevent the proposed deletion by removing the {{proposed deletion/dated}} notice, but please explain why in your edit summary or on the article's talk page.

Please consider improving the article to address the issues raised. Removing {{proposed deletion/dated}} will stop the proposed deletion process, but other deletion processes exist. The speedy deletion process can result in deletion without discussion, and articles for deletion allows discussion to reach consensus for deletion. --GaborPete (talk) 15:49, 10 November 2010 (UTC)

Advanced Weak Convergence Results[edit]

This article does not include any advanced weak convergence results, i.e. convergence in general metric spaces (for example Donsker's theorem links to this page, yet there is no mention of the type of convergence actually happening in this theorem). This should be included. --Steffen Grønneberg 17:27, 13 April 2006 (UTC)


Has anyone else seen the "convergence in probability" function being denoted as plim? I recently saw an old textbook as part of a CFA primer course that used this notation, but I have not had success googling for this notation anywhere. Can someone confirm that this is valid notation? If so, can we add it to this article? - DropDeadGorgias (talk) 17:48, Jun 22, 2004 (UTC)

---I can confirm this notation. My econometrics text (Kmenta, Jan. Elements of Econometrics: 2nd Ed. University of Michigan Press: Ann Arbor, 1987.) includes this notation.

- Seconded - see Page 47 of these course notes for example:

uniform integrability[edit]

good work! the article though lacks from further convergence implications when the r.v. are uniformly integrable.

Kudos to whoever expanded this article[edit]

very well written ,with all the examples, better than textbooks even.

Uniform convergence?[edit]

What's the view on adding a section on uniform convergence? Or should that be treated under stochastic processes? If so, perhaps a link to the relevant article would be worthwhile?--Steve Kroon 14:14, 14 February 2007 (UTC)


This article would be greatly improved if some examples of sequences of measures could be added which satisfy one convergence concept but not others.

convergence and the weak law of large numbers[edit]

The article says that convergence in distribution is the notion of convergence used in the weak law of large numbers, and that convergence in probability is the notion of convergence used in the weak law of large numbers. One of these statements should be false since the two notions of convergence are different, right? 09:39, 28 March 2007 (UTC)

The article says that convergence in distribution is the notion of convergence used in the weak law of large numbers,

I can't find anything in the article saying that. Can you point it out explicitly? Michael Hardy 22:43, 28 March 2007 (UTC)

Sorry for the delay, this is from the article:

Convergence in distribution is the weakest form of convergence, and is sometimes called weak convergence (main article: weak convergence of measures). It does not, in general, imply any other mode of convergence. However, convergence in distribution is implied by all other modes of convergence mentioned in this article, and hence, it is the most common and often the most useful form of convergence of random variables. It is the notion of convergence used in the central limit theorem and the (weak) law of large numbers.

I noticed the same thing -- which of the two statements is correct? --Lavaka 17:00, 18 May 2007 (UTC)
Both are correct. The expected value which the sample mean converges to is a constant, which gives us that convergence in probability and convergence in distribution are equivalent in this case.Aastrup 23:29, 24 July 2007 (UTC)

Convergence to random variables[edit]

This article seems to take for granted the difference between converging to a function (e.g., sure convergence and almost sure convergence) and converging to a random variable (e.g., the other forms of convergence). Note that you can almost surely convergence to a function that is not a random variable (i.e., not a Borel measurable function. It would be nice if this was cleared up. This would make the definitions make more sense. --TedPavlic 18:47, 8 April 2007 (UTC)

Convegence of NETS and General Stochastic Processes[edit]

All of the convergence ideas are in terms of SEQUENCES. That is, they all involve countable index sets. Convergence is defined for uncountable index sets in general. These definitions should be adjusted so that they refer to any general random process, regardless of the countability of its index set. --TedPavlic 18:47, 8 April 2007 (UTC)

Leap of illogic[edit]

In the proof that convergence in probability implies convergence in distribution, it's stated that:

But Pr(X ≤ a) is the cumulative distribution function FX(a), which is continuous by hypothesis, (...)

This hypothesis is not stated in the demonstration, and the cdf could be discontinuous. Albmont (talk) 17:22, 30 October 2008 (UTC)

I think one could consider the hypothesis to have been stated in the definition of convergence in distribution:
We say that the sequence Xn converges towards X in distribution, if
\lim_{n\rightarrow\infty} F_n(a) = F(a),\,
for every real number a at which F is continuous.
But the article could probably be more explicit about that. Michael Hardy (talk) 22:50, 30 October 2008 (UTC)
Ok, I think I overlooked it. Hmmm... this seems weird... but, on a second thought, the points where a càdlàg and monotonic function is discontinuous must be a set of zero measure (are they?), so ignoring them is not a big deal. BTW, the article could have some counterexamples, I will think about them and post here in the discussion. Albmont (talk) 10:44, 31 October 2008 (UTC)
OTOH, I think the wikibook might be a better place to include counter-examples. Albmont (talk) 10:51, 31 October 2008 (UTC)

Why they're ignored may not be the fact that their total measure is zero, but rather something else. Consider for example, a probability distribution that concentrates probability 1 at 1/n. As n grows, this approaches (in distribution) the "delta measure" that concentrates probability 1 at 0. But now observe the probability assigned to the set {0} by the nth measure in this sequence: it is 0, for all values of n. Yet in the limit it is 1. That's the point of discontinuity of the limiting c.d.f. Michael Hardy (talk) 15:35, 31 October 2008 (UTC)

I see... But I still find this forgiveness strange. Take, for example, a probability distribution that assigns probability 1/2 to zero and 1/2 to n. As n grows, this approaches (in distribution) the "delta measure" (!). Or did I just repeat some canonical counterexample of  (\lnot \xrightarrow{P}) \land \xrightarrow{\mathcal D}\,? :-) Albmont (talk) 16:45, 31 October 2008 (UTC)
Oops, just saw my error. Fn(1/2) would not converge to F(1/2). Albmont (talk) 16:48, 31 October 2008 (UTC)
In self-abasement for the nonsense I wrote :-), I copied (and adapted) your example to the wikibooks article. It still lacks counter-examples, and I still can't figure out a nice one. Albmont (talk) 17:14, 31 October 2008 (UTC)

Convergence to moving targets[edit]

The article definitions of convergence don't account for usage of convergence for two sequences of random variables, X_n and Y_n, so that X_n \xrightarrow{\mathcal D} Y_n, when neither limit is sensible as such. This is usually used for various ways of stating that the sum of suitable random variables converges to a normal distribution - as used in the article under "implications": "If Sn is a sum of n real independent random variables: S_n = X_1+\cdots+X_n then Sn converges almost surely if and only if Sn converges in probability." It's also used in the central limit theorem article: "the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound". It's fairly commonplace outside WP, too, so explaining it somewhere would be nice. My explanation above was maybe a bit overly general as I haven't seen this used for anything that rescaling won't fix (i.e. using mean instead of sum), but I think such statements are somewhat informal explanations of "convergence in shape of distribution". -- Coffee2theorems (talk) 18:26, 10 November 2008 (UTC)


I’m removing the statement that the following d(·,·) metrizes convergence in probability:

    d(X, Y) := \operatorname{E} \big[ \min \{|X-Y|, 1\} \big],

since the provided reference (Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3540520139. MR 1102015.  ; chapter 2) does not give any rationale for why is this indeed a metric and why is it equivalent to convergence in probability.

I replace the old metric with the Ky Fan metric (see the article), since the referenced text does prove the theorem that this metric metrizes convergence in probability. ... stpasha » talk » 20:42, 14 September 2009 (UTC)

Convergence almost surely: Example 3[edit]

I removed this example from the "convergence almost surely" section:

The primary reason is that f(i) is not a valid limiting variable, and thus we cannot say that Ui converges f(i). Of course one can claim that what is understood here is that Uif(i) converges a.s. to zero, but this is equivalent to saying that Xi converges a.s. to zero, which makes this example way too similar to the example 1. Besides, the statement “history bears out that all businesses sooner or later fold up” is dubious as well: there is at least one business, the Catholic church, which survived for over 1500 years already and doesn't show any signs of giving up.  … stpasha »  07:44, 21 November 2009 (UTC)

Relevance to encylopedia[edit]

A general complaint on ALL math, law, science, finance, sports or specialty articles; they are not written for the general reader. Rather we do our particular fields or professions a great disservice when we write only for ourselves. This is supposed to be a general purpose reader's encyclopedia. Check this out yourself and look at articles in physics or CAPM. You need to already be a finance PhD or MBA to read some of this type of stuff. This showing off does no help. —Preceding unsigned comment added by (talk) 03:05, 19 January 2010 (UTC)

Sorry, but general complaints of this sort will not help to improve the article. True, people writing these articles are probably already knowledgeable enough in this field that they cannot understand what exactly is unclear in the article. However let me assure you that no wikipedian writes with the purpose to impress the reader, only to convey the knowledge.
Bearing this in mind, let's skip the lead (because it hasn't been cleaned-up after merge yet), and look at the “convergence in probability”, “convergence in distribution” and “convergence almost surely” sections — what exactly you find confusing there, and how you might suggest the exposition could be improved?  // stpasha »  03:51, 17 February 2010 (UTC)

Archer example[edit]

The example for the archer is very good, however it is wrong ! The sequence converges in probability, but saying that it doesnt converge almost surely is false, it all depends on how fast the probability of not hitting the bullseye decreases ! For exemple, if the probability of hitting the bullseye for the n-th shot is (1 − 2n), then Borel-Cantelli assures you that it also converges almost surely. However, if the hit probabilty is (1 − 1/n) then it doesn't converge almost surely. —Preceding unsigned comment added by (talk) 09:40, 4 March 2010 (UTC) I too think the archer example is wrong for the reason given above. It should be removed. I'm going to have to tell my students not to look at this page. —Preceding unsigned comment added by (talk) 16:29, 25 February 2011 (UTC) I also agree - the example is simply wrong.

The example is correct, only underspecified. Although your remark is quite valid in the regards that if the probabilities were equal (1 − 2n) then the sequence would have converged almost surely as well as in probability, this only demonstrates that in reality those probabilities cannot converge to zero so rapidly. The example appeals to the everyday's experience of a lay-reader. We know that no archer is perfect. However if the probabilities were (1 − 2n) then we would be observing that at least some of the archers in the world have became perfect and never err anymore. Since this doesn’t happen, we can conclude that convergence almost surely doesn’t happen.  // stpasha »  00:47, 27 March 2010 (UTC)

I think both the example and this observation are great! Wikipedia maths articles and other maths literature would be so much richer if accompanied by such examples. —Preceding unsigned comment added by (talk) 12:23, 26 March 2010 (UTC)

Great they are, but for the wrong question. I believe the example and the two comments above are confusing the term "infinitely often" with "almost sure convergence". An event (or rather a sequence of events) is said to occur "infinitely often" if no matter how far down the sequence we go the event will at some point occur again. For example if the archers chance of missing is 1/n on his nth shot then although he is continually improving he will always have another miss to come at some point in the future. However this article is about modes of convergence, and the archer example doesn't say anything on this. Modes of convergence is to do with how the functions(random variables) map from the probability space to the values in the range of the function. To see the difference between "convergence almost surely" and "convergence in probability". Take your probability space to be the uniform measure on the [0,1] interval. Then take a function like f_n(x) = 1 if x is between [k/2^m,(k+1)/2^m] and 0 otherwise, where n= 2^m + k and k goes from 0 to 2^m -1. That is the sequence for (m,k) goes (0,0),(1,0),(1,1),(2,0),(2,1),(2,2),(2,3). This function converges in probability to f(x) = 0 but does not converge almost surely. This is because the "bump" is always moving across the probability space. So the probability P(f_n ≠ 0) = 1/2^m. However for any particular x in [0,1] f_n(x) does not converge to 0 (there is always a 1 in there from time to time). So the set of x in [0,1] where convergence holds is the empty set so we do not have almost sure convergence. Some graphics of the sequence of functions I've described would make for a good example. --Stryac (talk) 16:38, 1 July 2010 (UTC)

Possible expansion[edit]

On the "requests" page there has been a request for an article on "stochasic equicontinuity", referencing JSTOR 2938179 as a possible start. This also considers "uniform convergence in probability". One or both things might reasonably be dealt with by expanding this article, rather than having separate articles. 14:34, 14 September 2010 (UTC)

I believe this article is not the right place. Stochastic equicontinuity is the extension of the equicontinuity property to random variables (and probably should be discussed there). Stochastic equicontinuity is not a mode of convergence, but rather a technical condition needed to ensure that pointwise convergence in probability transforms into the uniform convergence in probability.  // stpasha »  16:27, 14 September 2010 (UTC)

Diagram of the relationship between convergence types[edit]

I think it would be helpful if there was a diagram showing the relationship between the different types of convergence. I have taken a stab at making one:

I'm happy to share the photoshop and mathematica file I used to make it with anyone who would like to improve on it. I think something like this diagram would be helpful in the article for people to quickly assimilate the information. —Preceding unsigned comment added by Emptiless (talkcontribs) 17:20, 8 October 2010 (UTC)

What about a simpler version like this: —Preceding unsigned comment added by Hsandsmark (talkcontribs) 20:48, 29 October 2010 (UTC)

Almost sure convergence vs. sure convergence[edit]

The examples provided in the "almost sure convergence" section are also examples of "sure convergence". I would move them to "sure convergence" examples. The main difference is that "almost sure convergence" allows for violations on a set of measure 0, and these examples don't highlight this. I would suggest putting an example that involves a set of measure 0. Off the top of my head, maybe something like this: consider a random variable x drawn uniformly on (0,1). Let C_n be a sequence that converges to the Cantor set. Then what is the probability that x is in C_n? Or, to make a random variable, the random variable y_n is 1 if x_n is in C_n, and 0 else. Then because the Cantor set has zero measure, y_n converges to 1 almost surely. Sorry this example isn't very precise -- maybe there's a much better one from standard textbooks. But my point is, I think the example should involve a zero-measure set. Lavaka (talk) 19:27, 2 December 2010 (UTC)

Uniformly in some parameter[edit]

for example Q_n(Y_n,theta) converges to Q(theta) almost surely uninformly in theta. How is this defined exactly??? Jackzhp (talk) 21:21, 13 January 2011 (UTC)

Graphic example of convergence on the right[edit]

The animation shows that for n=3,4,.. the distribution is smooth - makes no sense to me. — Preceding unsigned comment added by (talk) 14:23, 10 February 2012 (UTC)

Convergence in mean[edit]

I am removing the subsection on convergence in rth-mean because it replicates (badly) the section it belongs to! Also the example is kind of silly IMO, but if someone feels like it is useful then put it back by all means. (talk) 16:44, 10 September 2012 (UTC)

I added something that I think is useful at the end! — Preceding unsigned comment added by (talk) 18:04, 14 June 2014 (UTC)

Converges Almost Surely doesn't mean eventually constant[edit]

The examples seem to me to give the impression that in order to converge almost surely, the sequence must (with probability 1) become constant. But that's not right. It's just saying that with probability 1 the sequence converges -- in other words, you can pick a large enough N to ensure that X_n is no more than eps away from X, for any particular choice of eps. To compare to the archer example: the reason it converges in probability but doesn't converge almost surely is because we can pick some definition of a "bad miss", and the archer will still miss by that much every once in a while, but those misses become rarer and rarer, such that we can make them arbitrarily rare by taking the number of previous shots to be large enough. But we can't take the number of previous shots large enough to ensure he will never have a bad miss again. The probability of a bad miss goes to zero, but doesn't reach zero in some finite number of shots. In contrast, if for each definition of a "bad miss" we could pick some number of previous shots after which he would never have a bad miss again (i.e., the probability of another bad miss is zero), then it would converge almost surely. But that doesn't mean he never misses by *any* amount from then on. Rather, after some number of shots he will no longer have a "bad miss", after some greater number of shots he will no longer have a "somewhat bad miss", after some even greater number of shots he will no longer have a "just slightly bad miss", etc. - Tim314 (talk) 16:41, 25 June 2015 (UTC)