In the theory of discrete random matrices (e.g. matrices whose entries are random signs ), one often encounters the problem of understanding the distribution of the random variable , where is an -dimensional random sign vector (so is uniformly distributed in the discrete cube ), and is some -dimensional subspace of for some .

It is not hard to compute the second moment of this random variable. Indeed, if denotes the orthogonal projection matrix from to the orthogonal complement of , then one observes that

and so upon taking expectations we see that

since is a rank orthogonal projection. So we expect to be about on the average.

In fact, one has sharp concentration around this value, in the sense that with high probability. More precisely, we have

Proposition 1 (Large deviation inequality)For any , one hasfor some absolute constants .

In fact the constants are very civilised; for large one can basically take and , for instance. This type of concentration, particularly for subspaces of moderately large codimension , is fundamental to much of my work on random matrices with Van Vu, starting with our first paper (in which this proposition first appears). (For subspaces of small codimension (such as hyperplanes) one has to use other tools to get good results, such as inverse Littlewood-Offord theory or the Berry-Esséen central limit theorem, but that is another story.)

Proposition 1 is an easy consequence of the second moment computation and Talagrand’s inequality, which among other things provides a sharp concentration result for convex Lipschitz functions on the cube ; since is indeed a convex Lipschitz function, this inequality can be applied immediately. The proof of Talagrand’s inequality is short and can be found in several textbooks (e.g. Alon and Spencer), but I thought I would reproduce the argument here (specialised to the convex case), mostly to force myself to learn the proof properly. Note the concentration of obtained by Talagrand’s inequality is much stronger than what one would get from more elementary tools such as Azuma’s inequality or McDiarmid’s inequality, which would only give concentration of about or so (which is in fact trivial, since the cube has diameter ); the point is that Talagrand’s inequality is very effective at exploiting the convexity of the problem, as well as the Lipschitz nature of the function in all directions, whereas Azuma’s inequality can only easily take advantage of the Lipschitz nature of the function in coordinate directions. On the other hand, Azuma’s inequality works just as well if the metric is replaced with the larger metric, and one can conclude that the distance between and concentrates around its median to a width , which is a more non-trivial fact than the concentration bound given by that inequality. (The computation of the median of the distance is more complicated than for the distance, though, and depends on the orientation of .)

Remark 1If one makes the coordinates of iid Gaussian variables rather than random signs, then Proposition 1 is much easier to prove; the probability distribution of a Gaussian vector is rotation-invariant, so one can rotate to be, say, , at which point is clearly the sum of independent squares of Gaussians (i.e. a chi-square distribution), and the claim follows from direct computation (or one can use the Chernoff inequality). The gaussian counterpart of Talagrand’s inequality is more classical, being essentially due to Lévy, and will also be discussed later in this post.

** — 1. Concentration on the cube — **

Proposition 1 follows easily from the following statement, that asserts that if a convex set occupies a non-trivial fraction of the cube , then the neighbourhood will occupy almost all of the cube for :

Proposition 2 (Talagrand’s concentration inequality)Let be a convex set in . Thenfor all and some absolute constant , where is chosen uniformly from .

Remark 2It is crucial that is convex here. If instead is, say, the set of all points in with fewer than ‘s, then is comparable to , but only starts decaying once , rather than . Indeed, it is not hard to show that Proposition 2 implies the variantfor non-convex (by restricting to and then passing from to the convex hull, noting that distances to on may be contracted by a factor of by this latter process); this inequality can also be easily deduced from Azuma’s inequality.

To apply this proposition to the situation at hand, observe that if is the cylindrical region for some , then is convex and is contained in . Thus

Applying this with or , where is the median value of , one soon obtains concentration around the median:

This is only compatible with (1) if , and the claim follows.

To prove Proposition 2, we use the exponential moment method. Indeed, it suffices by Markov’s inequality to show that

for a sufficiently small absolute constant (in fact one can take ).

We prove (2) by an induction on the dimension . The claim is trivial for , so suppose and the claim has already been proven for .

Let us write for . For each , we introduce the slice , then is convex. We now try to bound the left-hand side of (2) in terms of rather than . Clearly

By symmetry we may assume that , thus we may write

Now we look at . For , let be the closest point of (the closure of) to , thus

Let be chosen later; then the point lies in by convexity, and so

Squaring this and using Pythagoras, one obtains

As we will shortly be exponentiating the left-hand side, we need to linearise the right-hand side. Accordingly, we will exploit the convexity of the function to bound

and thus by (4)

We exponentiate this and take expectations in (holding fixed for now) to get

Meanwhile, from the induction hypothesis and (3) we have

and similarly for . By Hölder’s inequality, we conclude

For , the optimal choice of here is , obtaining

for , the optimal choice of is to be determined. Averaging, we obtain

so to establish (2), it suffices to pick such that

If is bounded away from zero, then by choosing we would obtain the claim if is small enough, so we may take to be small. But then a Taylor expansion allows us to conclude if we take to be a constant multiple of , and again pick to be small enough. The point is that already almost works up to errors of , and increasing from zero to a small non-zero quantity will decrease the LHS by about . [By optimising everything using first-year calculus, one eventually gets the constant claimed earlier.]

Remark 3Talagrand’s inequality is in fact far more general than this; it applies to arbitrary products of probability spaces, rather than just to , and to non-convex , but the notion of distance needed to define becomes more complicated; the proof of the inequality, though, is essentially the same. Besides its applicability to convex Lipschitz functions, Talagrand’s inequality is also very useful for controlling combinatorial Lipschitz functions which are “locally certifiable” in the sense that whenever is larger than some threshold , then there exist some bounded number of coefficients of which “certify” this fact (in the sense that for any other which agrees with on these coefficients). See e.g. the text of Alon and Spencer for a more precise statement and some applications.

** — 2. Gaussian concentration — **

As mentioned earlier, there are analogous results when the uniform distribution on the cube are replaced by other distributions, such as the -dimensional Gaussian distribution. In fact, in this case convexity is not needed:

Proposition 3 (Gaussian concentration inequality)Let be a measurable set in . Thenfor all and some absolute constant , where is a random Gaussian vector.

This inequality can be deduced from Lévy’s classical concentration of measure inequality for the sphere (with the optimal constant), but we will give an alternate proof due to Maurey and Pisier. It suffices to prove the following variant of Proposition 3:

Proposition 4 (Gaussian concentration inequality for Lipschitz functions)Let be a function which is Lipschitz with constant (i.e. for all . Then for any we havefor all and some absolute constant , where is a random variable. [Informally, Lipschitz functions of Gaussian variables concentrate as if they were Gaussian themselves; for comparison, Talagrand’s inequality implies that

convexLipschitz functions ofBernoullivariables concentrate as if they were Gaussian.]

Indeed, if one sets , and splits into the cases whether or , one obtains either (in the former case) or , and so

in either case. Also, since , we have . Putting the two bounds together gives the claim.

Now we prove Proposition 4. By the epsilon regularisation argument we may take to be smooth, and so by the Lipschitz property we have

for all . By subtracting off the mean we may assume . By replacing with if necessary it suffices to control the upper tail probability for .

We again use the exponential moment method. It suffices to show that

for some absolute constant .

Now we use a variant of the square and rearrange trick. Let be an independent copy of . Since , we see from Jensen’s inequality that , and so

With an eye to exploiting (5), one might seek to use the fundamental theorem of calculus to write

But actually it turns out to be smarter to use a circular arc of integration, rather than a line segment:

The reason for this is that is another gaussian random variable equivalent to , as is its derivative ; furthermore, and crucially, these two random variables are *independent*.

To exploit this, we first use Jensen’s inequality to bound

Applying the chain rule and taking expectations, we have

Let us condition to be fixed, then ; applying (5), we conclude that is normally distributed with standard deviation at most . As such we have

for some absolute constant ; integrating out the conditioning on we obtain the claim.

## 38 comments

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9 June, 2009 at 8:39 pm

studentDear Prof Tao,

How do you pass from line segment integral to circular arc integral?

by change of variable ?

thanks

9 June, 2009 at 9:28 pm

Terence TaoDear student,

The circular arc identity is established directly from the fundamental theorem of calculus , rather than from applying a change of variables from the line segment identity (which doesn’t work). The point is that in higher dimensions, there are many inequivalent ways to apply the fundamental theorem of calculus to expand f(Y)-f(X) – one for each path from X to Y, and the key is to select the right one.

15 June, 2009 at 10:35 pm

studentDear Prof. Tao,

I remember once you wrote that you will teach random matrix theory next semester. my request is that

Would you please start to post little by little some notes related to prerequisite material for that course? if we start to study during summer and get enough background it would be perfect for us.

thank you very much

16 June, 2009 at 7:10 am

Terence TaoActually, the class will be starting in the winter quarter (next January). I have not yet decided exactly what the topics will be, but apart from a basic graduate education in analysis, probability, linear algebra, and combinatorics it should be fairly self-contained.

16 June, 2009 at 11:59 am

OdedOne minor comment: the statement in Remark 2 seems almost vacuous. After all, the distance between *any* two points in {-1,+1}^n is at most 2\sqrt{n}. Maybe the correct behavior should be n^{1/4}?

16 June, 2009 at 12:25 pm

Terence TaoHmm, good point. On the other hand, the Azuma/McDiarmid argument also works if the metric is replaced by the much larger metric (although the mean or median of is somewhat harder to compute in that case). I’ll adjust the text to reflect this. (One could make the case that Talagrand’s inequality is to as Azuma’s inequality or McDiarmid’s inequality is to ; note the crucial use of Pythagoras’ theorem in the middle of the proof of Talagrand’s inequality.)

16 June, 2009 at 8:58 pm

vedadiDear Prof. Tao,

just after ” square and rearrange trick”, you say that you use Jensen’s inequality.

I guess your convex function is for

in that case why do not we have ?

Jensen’s ineq is , right?

thanks

16 June, 2009 at 9:16 pm

Terence TaoDear vedadi,

Yes, that was a typo, the expectation should be at least 1, rather than at most 1. (The subsequent inequalities did not have the typo, though.)

17 June, 2009 at 9:11 pm

studentDear Prof Tao,

When we say ”let Y be an independent copy of X….” How do we guarantee the existence of that copy? Don’t we have some philosophical questions here too?

thanks

18 June, 2009 at 8:59 am

Terence TaoDear student,

Independent copies of a random variable can always be generated by the product measure construction,

http://en.wikipedia.org/wiki/Product_measure

This requires extending the underlying probability space to a finer one, but probability theory is designed in such a way that one can always do this without affecting any probabilistically meaningful quantities, such as probabilities of events or expectations, moments, correlations, and distributions of random variables. (As such, one of the guiding philosophies of probability theory is to not work with the underlying probability space if at all possible. The situation is somewhat analogous to differential geometry, which is designed to be invariant under changes of coordinates, and so the philosophy is often to avoid explicit manipulation of such coordinates whenever possible.)

Kallenberg’s “Foundations of modern probability” is a good reference for these sorts of foundational issues. However, it should be said that while such issues are of course important, especially from a philosophical or logical point of view, they rarely affect day-to-day usage of probability, much as the construction of the real numbers does not play a prominent role in real arithmetic, or as the construction of the Lebesgue integral does not play a prominent role in computation and estimation of integrals.

18 June, 2009 at 11:39 am

AnonymousDear Prof. Tao,

Could you please write a blog post on ”Large Deviation Principles” this summer by explaining the theory by examples and from your point of view?

thanks

2 July, 2009 at 6:05 am

ebrima sayeYou are inspirational!!

2 July, 2009 at 7:57 pm

studentDear Prof. Tao,

I could not find in which part I am doing a mistake in the following argument :

we know that if two random variables have the same distribution then their mean is the same. now

Let be a standard normal rv. and be an independent copy of it. then and have the same distribution (?) but and

what is going on here?

thanks

3 July, 2009 at 7:58 am

Terence TaoDear student:

The same issue crops up if Z is a Bernoulli distribution (equal to +1 with probability 1/2, and -1 with probability -1/2). This example is elementary enough that you can work out everything from first principles; if you do so, you should see the error.

Dear Ben: X decomposes into two orthogonal components, PX and X-PX. X-PX is the orthogonal projection of X to V, so the distance from X to V is the length of PX. Since P is an orthogonal projection, .

Alternatively, one can pick a coordinate system so that V is a coordinate plane, in which case all calculations can be done explicitly.

3 July, 2009 at 4:56 am

benDear Prof Tao

could you explain why

?

thanks

5 July, 2009 at 3:33 pm

AnonymousDear Prof. Tao,

I did not see at which step you are using the independence of and

and in the last paragraf, when you say ” let’s condition to be fixed….” do you mean you that you are using the following fact:

thanks

6 July, 2009 at 12:57 pm

Terence TaoYes, this is what is happening when one is conditioning on . The independence of and is used precisely at this step, to ensure that retains its normal distribution even after conditioning on a fixed value of .

6 July, 2009 at 11:19 pm

lutfuDear Prof. Tao,

Let be two independent standart normal r.vs. then

since we have

therefore

is this argument true?

in which step we are using independence of and

thanks

7 July, 2009 at 6:33 am

Terence TaoYes, this is correct. The independence is needed to establish the identity (because one needs X to remain normally distributed after conditioning on the event Y=t).

8 July, 2009 at 4:07 pm

lutfuDear Prof Tao,

Let be a seq of i.i.d. rvs. with mean zero.

as we know from as the distribution of

concentrates on a point, i.e. limiting distribution is (a.s. implies in distribution)

Under the existence of second moment we know the limiting distribution of

is standard normal, i.e Gaussian measure.

my question is that do we know the distribution of

for under the suitable conditions? to which measure do they concentrate?

thanks

8 July, 2009 at 7:57 pm

AnonymousDear Prof. Tao,

Could you please give a proof for the Azuma-Hoeffding inequality?

thanks

3 January, 2010 at 10:22 pm

254A, Notes 1: Concentration of measure « What’s new[…] by induction on dimension . In the case when are Bernoulli variables, this can be done; see this previous blog post. In the general case, it turns out that in order to close the induction properly, one must […]

5 January, 2010 at 4:20 pm

254A, Notes 0: A review of probability theory « What’s new[…] by induction on dimension . In the case when are Bernoulli variables, this can be done; see this previous blog post. In the general case, it turns out that in order to close the induction properly, one must […]

13 June, 2010 at 1:23 pm

Laurent JacquesDear Terence,

Thank you for this verrry interesting post. I’d like to share with you some elements.

From what I know, Sub-Gaussian Random Variable (SGRV), such as Bernoulli RV, centered Uniform RV, Gaussian RV, … are defined as follows:

X is a SGRV if there exists one c>0 such that for all real value s,

E[ \exp (s X) ] <= exp (c^2 s^2/2)

See for instance V. V. Buldygin publications, e.g. "Sub-Gaussian random variables", doi:10.1007/BF01087176

If I'm not wrong, one interesting property of SBRV is that any linear combination of SGRV is SGRV with known characteristics. In particular, the "Gaussian standard", defined as the minimal "c" such that the definition above holds, behaves as the standard deviation of Gaussian RV with respect to linear combination of SGRV.

This implies also that if you rotate a sub-Gaussian random vector with iid SGRV components, you get also a sub-Gaussian random vector. The components of this latter are perhaps not iid anymore but at least they have all the same "Gaussian standard".

My question is thus the following:

**Do you think that Theorem 4 above could be easily generalized to sub-Gaussian random vectors of components with unit gaussian standard?**

It seems indeed that in many places in you proof, you use already key ingredients for this generalization:

* the sub-gaussian inequality definition;

* linear combinations (rotation) of Gaussian RVs which are themselves Gaussian.

Or perhaps this generalization is already known in the community (I didn't see it anyway in Talagrand/Ledoux's books)?

Best,

Laurent

18 August, 2010 at 12:43 am

AnonymousDear Prof. Tao

I have two questions about the proof for Proposition 4.

Firstly, It seems that there is a typo when you bound

exp(t(F(X)-F(Y))) by using Jensen Inequality, please check it if possible.

Secondly, the aim is

Eexp(tF(X))<=exp(Ct^2),

but the proof only tells us

Eexp(tF(X))<=exp(Ct),

Please give some explaination if possible.

Thanks.

[Corrected, thanks – T.]20 August, 2010 at 2:50 am

AnonymousDear Prof. Tao

Have you checked the bound of exp(t(F(X)-F(Y))) in the proof of Proposition 4? There is still a typo in my opinion.

[Corrected, thanks – T.]21 August, 2010 at 2:16 am

AnonymousDear Prof. Tao

Can you tell me how to get ? Some hints will be OK.

21 August, 2010 at 7:30 am

Terence TaoIf X is a Gaussian with mean zero and variance , then the moment generating function is , as can be seen by completing the square.

22 August, 2010 at 4:41 am

AnonymousDear Prof. Tao

I get it. Thanks for your reply. So far I have got a lot of interesting and useful mathemetical knowledge from your blog. Your earnest attitude and diligence impressed me. Thanks again.

16 November, 2010 at 10:04 pm

The Determinant of Random Bernoulli Matrices « Tsourolampis Blog[…] [6] Talagrand’s inequality http://terrytao.wordpress.com/2009/06/09/talagrands-concentration-inequality/ […]

25 April, 2011 at 2:44 pm

AnonymousDear Prof. Tao,

is Talagrand concentration inequality still valid if we have independent but not identically distributed normally distributed component of ?

Thanks

8 May, 2012 at 1:15 am

AnonymousHi Professor Tao,

Does dist(X,V) = euclidean distance(X,V)?

Thanks.

[Yes – T.]12 May, 2012 at 5:32 pm

sabyasachi chatterjeeHello Professor Tao,

I have had to investigate the concentration of the log determinant of a matrix. Specifically, suppose M1,..Mk are random matrices p by p, iid from a given distribution(can assume that the entries of the matrix are bounded a.s or other regularity conds if needed) whose expected value is the identity matrix I. Now if I consider the matrix mean..(M1+…Mk)/k and look at its log determinant..will it concentrate around its mean? Will the mean go to zero as k go to infinity? Hence will it concentrate around 0? Is there any known result on this? Thanks

9 July, 2012 at 10:53 pm

mathepsilonDoes that mean, the small constant c in the gaussian concentration inequaltiy for lipschitz function is (pi^2)/8? Thank you! I really can not find this constant elsewhere.

10 July, 2012 at 9:08 am

Terence TaoWell, the arguments given in the above post are not necessarily optimal; there may be other arguments that give a better value of c. (Note also that the value of c in Proposition 3 and Proposition 4 may differ, for instance the above arguments give a loss of a factor of 4 when passing from the latter to the former.)

I would try Ledoux’s book, as it has a number of proofs of this inequality and may discuss the question of optimal constants somewhere. (For instance, log-Sobolev methods tend to be quite sharp in general, and should yield good results in this case.)

17 December, 2013 at 2:11 pm

AnonymousDear Terry Tao,

can you get limit behavior of dist(X,V) (using Talagrand inequality or otherwise)?

Proposition 1 suggests that perhaps (after properly normalizing), the distribution of dist(X,V) converges to the normal distribution. Is it true?

I think it would nicely illustrate the strength of the presented method — similarly to the comparison of Chernoff bound and Central limit theorem.

Thanks,

Robert Samal

17 December, 2013 at 4:42 pm

Terence TaoWhen X is a gaussian vector, then dist(X,V)^2 is a chi-squared distribution (with the number of degrees of freedom equal to the codimension of V), and so the central limit theorem shows that dist(X,V)^2 (and thus also dist(X,V)) is approximately gaussian when the codimension is large. For small codimension, one gets a chi distribution instead.

In the non-gaussian case, one can presumably get similar results either through a Lindeberg exchange argument, or through a central limit theorem for quadratic forms, although I don’t know if this has been done explicitly in the literature.

28 August, 2014 at 11:39 pm

AnonymousIs proposition 4 valid for idd rvs with exponential moments?