1 Functions of normal random variables

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1 Tel Aviv University, 200 Gaussian measures : results formulated Functions of normal random variables It is of course impossible to even think the word Gaussian without immediately mentioning the most important property of Gaussian processes, that is concentration of measure. M. Talagrand a Gaussian isoperimetry and related inequalities. b Application: decoding error probability c Useful special cases d Application: frozen disorder models a Gaussian isoperimetry and related inequalities a Definition. (a) The standard Gaussian measure γ on R, called also the standard normal distribution N(0, ), is the probability measure γ (A) = e x2 /2 dx for Lebesgue measurable A R. 2π A (b) The standard Gaussian measure γ n on R n, called also the standard multinormal distribution, is the probability measure γ γ, that is, γ n (A) = (2π) n/2 e x 2 /2 dx for Lebesgue measurable A R n. A Here and below, is the Euclidean norm. A random vector (X,...,X n ) = X : Ω R n is distributed γ n if and only if X,...,X n are independent N(0, ) random variables. Let µ be a nonatomic probability measure on R, and ν an arbitrary probability measure on R. Then there exists an increasing f : R R such that f[µ] = ν, that is, f sends µ into ν. Such f is unique up to the values at points of discontinuity; it is unique if the support of ν is connected, which holds in all cases treated below. In particular, for every random variable ξ : Ω R there exists an increasing f : R R such that f(ζ) is distributed like ξ if ζ is distributed N(0, ). See page 89 of Mean field model for spin glasses: a first course, Lecture Notes in Math. 86 (2003),

2 Tel Aviv University, 200 Gaussian measures : results formulated 2 a2 Theorem. Let a function ξ : R n R satisfy the Lipschitz condition with constant : ξ Lip(), that is, x, y R n ξ(x) ξ(y) x y. Then ξ[γ n ] = f[γ ] for an increasing f : R R, f Lip(). The same holds for Lip(C) (by rescaling). Note that both the supremum and the infimum of Lip() functions is a Lip() function (if finite). a3 Theorem. 2 Let a function ξ : R n R be convex: x, y R n c [0, ] ξ(cx + ( c)y) cξ(x) + ( c)ξ(y). Then ξ[γ n ] = f[γ ] for a convex increasing f : R R. If ξ is Lip(C) and convex then f is (increasing and) Lip(C) and convex. In particular, this holds for a (semi)norm: ξ(x) = sup x, y y Y for a bounded Y R n ; in this case C = sup y Y y. In addition, x R f(x) Cx (for a seminorm). However, for a seminorm ξ it is natural to introduce an increasing g : (0, ) (0, ) such that ξ is distributed like g( ζ ), ζ sin N(0, ). It appears that the function a g(a) is decreasing on (0, ), a which is called the S-inequality. 3 Another result for a seminorm ξ may be formulated in terms of a random variable η Exp(), that is, a [0, ) P ( η a ) = e a, and a decreasing function h : (0, ) (0, ) such that ξ h(η). It appears that the function ln h( ) is convex on (0, ), which is called the B-inequality. 4 It is conjectured that for all seminorms ξ, ξ 2, P ( ξ and ξ 2 ) P ( ξ ) P ( ξ 2 ). This correlation conjecture is a 40-years old open problem! 5 V. Sudakov, B. Tsirelson 974, and independently C. Borell A. Ehrhard R. Latala, K. Oleszkiewicz D. Cordero-Erausquin, M. Fradelizi, B. Maurey See: G. Schechtman, Th. Schlumprecht, J. Zinn (998), On the Gaussian measure of the intersection, Annals of Probability 26:,

3 Tel Aviv University, 200 Gaussian measures : results formulated 3 b Application: decoding error probability A code of length n is a finite set C R n ; each c C is a codevector. A sender submits a given codevector c 0 to the additive white Gaussian noise channel, and it gets corrupted to c 0 + σx where X is distributed γ n and σ is the channel noise value. A receiver chooses the codevector c closest to the received vector c 0 + σx (which means maximum-likelihood decoding). Consider the error probability P error (σ) = P ( c c 0 ) and the minimal distance (for given C and c 0 ). D = min c C,c c 0 c c 0 b Theorem. There exists an increasing Lip() convex function f : R (0, ) such that where ζ N(0, ). ( σ > 0 P error (σ) = P f(ζ) D ) 2σ Defining the critical noise value σ c by P error (σ c ) = 0.5 we get ( σ (0, σ c ) P error (σ) P ζ D ( 2 σ )). σ c c Useful special cases Let y,...,y n R d, k y k. The function (c) ξ(x) = max k x, y k is Lip() and convex. The function (c2) ξ(x) = max x, y k k This is Theorem of: J.-P. Tillich, G. Zemor (2004) The Gaussian isoperimetric inequality and decoding error probabilities for the Gaussian channel, IEEE Transactions on Information Theory 50:2,

4 Tel Aviv University, 200 Gaussian measures : results formulated 4 is Lip() and a seminorm. For p [, ) the function ( ) /p (c3) ξ(x) = x, y k p n is Lip() and a seminorm. For c R, c 0, the function k (c4) ξ(x) = c ln k e c x,y k is Lip() and convex. More generally, for a nonempty subset A of the unit ball of R d and a probability measure µ on the unit ball of R d we may generalize (c) to ξ(x) = sup y A x, y, (c2) to ξ(x) = sup y A x, y, (c3) to ξ(x) = ( x, y p µ(dy) ) /p, and (c4) to ξ(x) = c ln e c x,y µ(dy). For every nonempty A R d the function (c5) ξ(x) = inf x y y A is Lip(). If A is convex then the function is also convex. d Application: frozen disorder models Let X k,l for 0 k m and k l k be independent N(0, ) random variables. Every path L = (l 0,...,l m ) such that l 0 = 0 and l k+ l k = ± (for k = 0,..., m ) leads to a random variable X L = m X k,lk. k=0 First-time percolation The random variable ξ m = max m L X L is a Lip() convex function of (X k,l ) k,l. Gaussian concentration helps to prove the strict inequality Eξ m < 2 ln2, m lim m

5 Tel Aviv University, 200 Gaussian measures : results formulated 5 and the same for d-dimensional l, with 2 ln(2d) in the right-hand side. For β > 0 the random variable Directed polymer Z m,β = 2 m L e βx L is the so-called partition function of the directed polymer in Gaussian random environment; β is the inverse temperature. (The energy of L is, by definition, X L ; the probability of L, given (X k,l ) k,l, is Z m,β e βx L.) The 2 m limit of E ln Z m m,β as m is of interest; 2 in dimension 3 it is equal to β 2 /2 for β below a critical value, but strictly less than β 2 /2 for β above the critical value. The random variable ξ = β m ln Z m,β is a Lip() convex function of (X k,l ) k,l. Spin glass Let X k,l for k < l m be independent N(0, ) random variables. Every (σ,..., σ m ) = σ {, +} m leads to a random variable X σ = k<l σ k σ l X k,l. For β > 0 the random variable Z m,β = 2 m σ e βxσ/ m is the partition function of the so-called Sherrington-Kirkpatrick model for spin glasses ( SK model, nonlocal; the energy of σ is, by definition, X σ / n; the probability of σ, given (X k,l ) k,l, is Z m,β 2 m e βxσ/ m ). The limit of m E ln Z m,β (as m ) is of interest; 3 it is equal to β 2 /4 for β <, but strictly less than β 2 /4 for β >. 4 See Theorem.2 in: Y. Hu, Q.-M. Shao (2009), A note on directed polymers in Gaussian environments, Electronic Communications in Probability 4, See Hu and Shao, Sect... 3 See Talagrand, page Rather, limsup(... ) < β 2 /4.

6 Tel Aviv University, 200 Gaussian measures : results formulated 6 The random variable ξ = β 2 m ln Z m,β is a Lip() convex function of (X k,l ) k,l. Gaussian concentration shows that m ln Z m,β is usually close to its expectation. This random function of β is convex and has small fluctuations. The quantity of information that can be extracted from this simple fact is amazing. (Talagrand, page 9) References [] V.I. Bogachev, Gaussian measures, AMS 998. [2] R. Latala, On some inequalities for Gaussian measures, Proc. Intern. Congress of Mathematicians (ICM 2002), vol. II,