Normal Distribution
characterizations with applications

W odzimierz Bryc
Department of Mathematics
University of Cincinnati
P O Box 210025
Cincinnati, OH 45221-0025
bryc@ucbeh.san.uc.edu

December 22, 1994

Preface

This book is a concise presentation of the normal distribution on the real line and its counterparts on more abstract spaces, which we shall call the Gaussian distributions. The material is selected towards presenting characteristic properties, or characterizations, of the normal distribution. There are many such properties and there are numerous relevant works in the literature. In this book special attention is given to characterizations generated by the so called Maxwell's Theorem of statistical mechanics, which is stated in the introduction as Theorem . These characterizations are of interest both intrinsically, and as techniques that are worth being aware of. The book may also serve as a good introduction to diverse analytic methods of probability theory. We use characteristic functions, tail estimates, and occasionally dive into complex analysis.

In the book we also show how the characteristic properties can be used to prove important results about the Gaussian processes and the abstract Gaussian vectors. For instance, in Section we present Fernique's beautiful proofs of the zero-one law and of the integrability of abstract Gaussian vectors. The central limit theorem is obtained via characterizations in Section .

The excellent book by Kagan, Linnik & Rao [] overlaps with ours in the coverage of the classical characterization results. Our presentation of these is sometimes less general, but in return we often give simpler proofs. On the other hand, we are more selective in the choice of characterizations we want to present, and we also point out some applications. Characterization results that are not included in [] can be found in numerous places of the book, see Section , Chapter and Chapter .

We have tried to make this book accessible to readers with various backgrounds. If possible, we give elementary proofs of important theorems, even if they are special cases of more advanced results. Proofs of several difficult classic results have been simplified. We have managed to avoid functional equations for non-differentiable functions; in many proofs in the literature lack of differentiability is a major technical difficulty.

The book is primarily aimed at graduate students in mathematical statistics and probability theory who would like to expand their bag of tools, to understand the inner workings of the normal distribution, and to explore the connections with other fields. Characterization aspects sometimes show up in unexpected places, cf. Diaconis & Ylvisaker []. More generally, when fitting any statistical model to the data, it is inevitable to refer to relevant properties of the population in question; otherwise several different models may fit the same set of empirical data, cf. W. Feller []. Monograph [] by Prakasa Rao is written from such perspective and for a statistician our book may only serve as a complementary source. On the other hand results presented in Sections and are quite recent and virtually unknown among statisticians. Their modeling aspects remain to be explored, see Section . We hope that this book will popularize the interesting and difficult area of conditional moment descriptions of random fields. Of course it is possible that such characterizations will finally end up far from real life like many other branches of applied mathematics. It is up to the readers of this book to see if the following sentence applies to characterizations as well as to trigonometric series.

``Thinking of the extent and refinement reached by the theory of trigonometric series in its long development one sometimes wonders why only relatively few of these advanced achievements find an application.''
(A. Zygmund, Trigonometric Series, Vol. 1, Cambridge Univ. Press, Second Edition, 1959, page xii)

There is more than one way to use this book. Parts of it have been used in a graduate one-quarter course Topics in statistics. The reader may also skim through it to find results that he needs; or look up the techniques that might be useful in his own research. The author of this book would be most happy if the reader treats this book as an adventure into the unknown - picks a piece of his liking and follows through and beyond the references. With this is mind, the book has a number of references and digressions. We have tried to point out the historical perspective, but also to get close to current research.

An appropriate background for reading the book is a one year course in real analysis including measure theory and abstract normed spaces, and a one-year course in complex analysis. Familiarity with conditional expectations would also help. Topics from probability theory are reviewed in Chapter , frequently with proofs and exercises. Exercise problems are at the end of the chapters; solutions or hints are in Appendix .

The book benefited from the comments of Chris Burdzy, Abram Kagan, Samuel Kotz, Wodek Smole\'nski, Pawe Szabowski, and Jacek Wesoowski. They read portions of the first draft, generously shared their criticism, and pointed out relevant references and errors. My colleagues at the University of Cincinnati also provided comments, criticism and encouragement. The final version of the book was prepared at the Institute for Applied Mathematics of the University of Minnesota in fall quarter of 1993 and at the Center for Stochastic Processes in Chapel Hill in Spring 1994. Support by C. P. Taft Memorial Fund in the summer of 1987 and in the spring of 1994 helped to begin and to conclude this endeavor.

Introduction

The following narrative comes from J. F. W. Herschel [].

``Suppose a ball is dropped from a given height, with the intention that it shall fall on a given mark. Fall as it may, its deviation from the mark is error, and the probability of that error is the unknown function of its square, ie. of the sum of the squares of its deviations in any two rectangular directions. Now, the probability of any deviation depending solely on its magnitude, and not on its direction, it follows that the probability of each of these rectangular deviations must be the same function of its square. And since the observed oblique deviation is equivalent to the two rectangular ones, supposed concurrent, and which are essentially independent of one another, and is, therefore, a compound event of which they are the simple independent constituents, therefore its probability will be the product of their separate probabilities. Thus the form of our unknown function comes to be determined from this condition...''

Ten years after Herschel, the reasoning was repeated by J. C. Maxwell []. In his theory of gases he assumed that gas consists of small elastic spheres bumping each other; this led to intricate mechanical considerations to analyze the velocities before and after the encounters. However, Maxwell answered the question of his Proposition IV: What is the distribution of velocities of the gas particles? without using the details of the interaction between the particles; it lead to the emergence of the trivariate normal distribution. The result that velocities are normally distributed is sometimes called Maxwell's theorem. At the time of discovery, probability theory was in its beginnings and the proof was considered ``controversial" by leading mathematicians.

The beauty of the reasoning lies in the fact that the interplay of two very natural assumptions: of independence and of rotation invariance, gives rise to the normal law of errors - the most important distribution in statistics. This interplay of independence and invariance shows up in many of the theorems presented below.

Here we state the Herschel-Maxwell theorem in modern notation but without proof; for one of the early proofs, see []. The reader will see several proofs that use various, usually weaker, assumptions in Theorems , , , , and .

Theorem 1 Suppose random variables X, Y have joint probability distribution m(dx, dy) such that

(i) m(·) is invariant under the rotations of IR²;

(ii) X, Y are independent.

Then X, Y are normally distributed.

This theorem has generated a vast literature. Here is a quick preview of pertinent results in this book.

Polya's theorem [] presented in Section says that if just two rotations by angles p/2 and p/4, preserve the distribution of X, then the distribution is normal. Generalizations to characterizations by the equality of distributions of more general linear forms are given in Chapter . One of the most interesting results here is Marcinkiewicz's theorem [], see Theorem .

An interesting modification of Theorem *, discovered by M. Sh. Braverman [] and presented in Section below, considers three i. i. d. random variables X, Y, Z with the rotation-invariance assumption (i) replaced by the requirement that only some absolute moments are rotation invariant.

Another insight is obtained, if one notices that assumption (i) of Maxwell's theorem implies that rotations preserve the independence of the original random variables X, Y. In this approach we consider a pair X, Y of independent random variables such that the rotation by an angle a produces two independent random variables Xcosa+Ysina and Xsina-Ycosa. Assuming this for all angles a, M. Kac [] showed that the distribution in question has to be normal. Moreover, careful inspection of Kac's proof reveals that the only essential property he had used was that X, Y are independent and that just one p/4-rotation: (X+Y)/ Ö2, (X-Y)/ Ö2 produces the independent pair. The result explicitly assuming the latter was found independently by Bernstein []. Bernstein's theorem and its extensions are considered in Chapter ; Bernstein's theorem also motivates the assumptions in Chapter .

 The following is a more technical description the contents of the book. Chapter collects probabilistic prerequisites. The emphasis is on analytic aspects; in particular elementary but useful tail estimates collected in Section . In Chapter we approach multivariate normal distributions through characteristic functions. This is a less intuitive but powerful method. It leads rapidly to several fundamental facts, and to associated Reproducing Kernel Hilbert Spaces (RKHS). As an illustration, we prove the large deviation estimates on IR^d which use the conjugate RKHS norm. In Chapter the reader is introduced to stability and equidistribution of linear forms in independent random variables. Stability is directly related to the CLT. We show that in the abstract setup stability is also responsible for the zero-one law. Chapter presents the analysis of rotation invariant distributions on IR^d and on IR^¥ . We study when a rotation invariant distribution has to be normal. In the process we analyze structural properties of rotation invariant laws and introduce the relevant techniques. In this chapter we also present surprising results on rotation invariance of the absolute moments. We conclude with a short proof of de Finetti's theorem and point out its implications for infinite spherically symmetric sequences. Chapter parallels Chapter in analyzing the role of independence of linear forms. We show that independence of certain linear forms, a characteristic property of the normal distribution, leads to the zero-one law, and it is also responsible for exponential moments. Chapter is a short introduction to measures of dependence and stability issues. Theorem establishes integrability under conditions of interest, eg. in polynomial biorthogonality as studied by Lancaster []. In Chapter we extend results in Chapter to conditional moments. Three interesting aspects emerge here. First, normality can frequently be recognized from the conditional moments of linear combinations of independent random variables; we illustrate this by a simple proof of the well known fact that the independence of the sample mean and the sample variance characterizes normal populations, and by the proof of the central limit theorem. Secondly, we show that for infinite sequences, conditional moments determine normality without any reference to independence. This part has its natural continuation in Chapter . Thirdly, in the exercises we point out the versatility of conditional moments in handling other infinitely divisible distributions. Chapter is a short introduction to continuous parameter random fields, analyzed through their conditional moments. We also present a self-contained analytic construction of the Wiener process.

Chapter 1
Probability tools

Most of the contents of this section is fairly standard probability theory. The reader shouldn't be under the impression that this chapter is a substitute for a systematic course in probability theory. We will skip important topics such as limit theorems. The emphasis here is on analytic methods; in particular characteristic functions will be extensively used throughout.

Let (W, M, P) be the probability space, ie. W is a set, M is a s-field of its subsets and P is the probability measure on (W, M). We follow the usual conventions: X,Y,Z stand for real random variables; boldface X, Y, Z denote vector-valued random variables. Throughout the book EX = ò_W X(w) dP (Lebesgue integral) denotes the expected value of a random variable X. We write X @ Y to denote the equality of distributions, ie. P(X Î A) = P(Y Î A) for all measurable sets A. Equalities and inequalities between random variables are to be interpreted almost surely (a. s.). For instance X £ Y+1 means P(X £ Y+1) = 1; the latter is a shortcut that we use for the expression P({w Î W: X(w) £ Y(w)+1}) = 1.

Boldface A, B, C will denote matrices. For a complex z = x+iy Î \sf CC by x = Âz and y = Áz we denote the real and the imaginary part of z. Unless otherwise stated, loga = log_ea denotes the natural logarithm of number a.

1.1  Moments

Given a real number r ³ 0, the absolute moment of order r is defined by E|X|^r; the ordinary moment of order r = 0, 1, ¼ is defined as EX^r. Clearly, not every sequence of numbers is the sequence of moments of a random variable X; it may also happen that two random variables with different distributions have the same moments. However, in Corollary below we will show that the latter cannot happen for normal distributions.

The following inequality is known as Chebyshev's inequality. Despite its simplicity it has numerous non-trivial applications, see eg. Theorem or [].

[ 1 If f: IR₊® IR₊ is a non-decreasing function and Ef(|X|) = C < ¥, then for all t > 0 such that f(t) ¹ 0 we have

Indeed, Ef(|X|) = ò_W f(|X|) dP ³ ò_{|X| ³ t}f(|X|) dP ³ ò_{|X| ³ t}f(t) dP = f(t)P(|X| > t).

It follows immediately from Chebyshev's inequality that if E|X|^p = C < ¥, then P(|X| > t) £ C/t^p, t > 0. An implication in converse direction is also well known: if P(|X| > t) £ C/t^p+e for some e > 0 and for all t > 0, then E|X|^p < ¥, see () below.

The following formula will often be useful¹.

[ 2 If f: IR₊® IR is a function such that f(x) = f(0)+ ò₀^xg(t) dt, E{|f(X)|} < ¥ and X ³ 0, then

Proof. The formula follows from Fubini's theorem², since for X ³ 0

[ 1 If E|X|^r < ¥ for an integer r > 0, then

Proof. Formula (4) follows directly from Proposition 1.1 (with f(x) = x^r and g(t) = [d/ dt]f(t) = rt^r-1).

Since EX = EX⁺ - EX^-, where X⁺ = max{X, 0} and X^- = min{X, 0}, therefore applying Proposition 1.1 separately to each of this expectations we get (3). ^[¯]

1.2  L_p-spaces

Several useful inequalities are collected in the following.

Theorem 2

• [(i)]for 1 £ p £ q £ ¥ we have Minkowski's inequality

  ||X||p £ ||X||q.

  (5)

• [(ii)]for 1/p+1/q = 1, p ³ 1 we have Hölder's inequality

  EXY £ ||X||p||Y||q.

  (6)

• [(iii)] for 1 £ p £ ¥ we have triangle inequality

  ||X+Y||p £ ||X||p+||Y||p.

  (7)

Special case p = q = 2 of Hölder's inequality (6) reads EXY £ [Ö(EX²EY²)]. It is frequently used and is known as the Cauchy-Schwarz inequality.

For 1 £ p < ¥ the conjugate space to L_p (ie. the space of all bounded linear functionals on L_p) is usually identified with L_q, where 1/p+1/q = 1. The identification is by the duality áf,gñ = òf(w)g(w) dP.

For the proof of Theorem 1.2 we need the following elementary inequality.

[ 1 For a,b > 0, 1 < p < ¥ and 1/p+1/q = 1 we have

Proof. Function t® t^p/p+t^-q/q has the derivative t^p-1-t^-q-1. The derivative is positive for t > 1 and negative for 0 < t < 1. Hence the maximum value of the function for t > 0 is attained at t = 1, giving

By Var(X) we shall denote the variance of a square integrable r. v. X

1.3  Tail estimates

Theorem 3 If there are C > 1, 0 < q < 1, x₀ ³ 0 such that for all x > x₀

Proof. Let a_n be such that when a_n = x_n-x₀ then a_n+1 = Cx_n. Solving the resulting recurrence we get a_n = Cⁿ-b, where b = Cx₀(C-1)^-1. Equation (9) says N(a_n+1) £ CN(a_n). Therefore

[ 2 If there is 0 < q < 1 and x₀ ³ 0 such that N(2x) £ q N(x-x₀) for all x > x₀, then E|X|^b < ¥ for all b < log₂ 1/q.

[ 3 Suppose there is C > 1 such that for every 0 < q < 1 one can find x₀ ³ 0 such that

As a special case of Corollary 1.3 we have the following.

[ 4 Suppose there are C > 1, K < ¥ such that

The next result deals with exponentially small tails.

Theorem 4 If there are C > 1, 1 < K < ¥, x₀ ³ 0 such that

Proof. As in the proof of Theorem 1.3, let a_n = Cⁿ-b, b = Cx₀/(C-1). Put q_n = log_KN(a_n). Then (12) gives

Since a_n®¥, we have N(a_n)® 0 and q_n®-¥. Choose m large enough to have 1+q_m < 0. Then (14) implies

^[¯]

[ 5 If there are C < ¥, x₀ ³ 0 such that

[ 6 If there are C < ¥, x₀ ³ 0 such that

1.4  Conditional expectations

Below we recall the definition of the conditional expectation of a r. v. with respect to a s-field and we state several results that we need for future reference. The definition is as old as axiomatic probability theory itself, see []. The reader not familiar with conditional expectations should consult textbooks, eg. Billingsley [], Durrett [], or Neveu [].

Definition 1 Let (W, M, P) be a probability space. If F Ì M is a s-field and X is an integrable random variable, then the conditional expectation of X given F is an integrable F-measurable random variable Z such that ò_AX dP = ò_A Z dP for all A Î F.

Conditional expectation of an integrable random variable X with respect to a s-field F Ì M will be denoted interchangeably by E{X| F} and E ^FX. We shall also write E{X|Y} or E^YX for the conditional expectation E{X| F} when F = s(Y) is the s-field generated by a random variable Y.

Existence and almost sure uniqueness of the conditional expectation E{X| F} follows from the Radon-Nikodym theorem, applied to the finite signed measures m(A) = ò_AX dP and P_{| F}, both defined on the measurable space (W, F). In some simple situations more explicit expressions can also be found.

Example. Suppose F is a s-field generated by the events A₁, A₂, ¼, A_n which form a non-degenerate disjoint partition of the probability space W. Then it is easy to check that

Example. Suppose that f(x, y) is the joint density with respect to the Lebesgue measure on IR² of the bivariate random variable (X, Y) and let f_Y(y) ¹ 0 be the (marginal) density of Y. Put f(x|y) = f(x, y)/f_Y(y). Then E{X|Y} = h(Y), where h(y) = ò_-¥^¥ x f(x|y) dx.

The next theorem lists properties of conditional expectations that will be used without further mention.

Theorem 5

• [(i)] If Y is F-measurable random variable such that X and XY are integrable, then E{XY| F} = YE{X| F};

• [(ii)] If G Ì F, then E ^G E ^F = E ^G;

• [(iii)] If s(X, F) and N are independent s-fields, then E{X| NÚ F} = E{X| F}; here NÚ F denotes the s-field generated by the union NÈ F;

• [(iv)] If g(x) is a convex function and E|g(X)| < ¥, then g(E{X| F}) £ E{g(X)| F};

• [(v)] If F is the trivial s-field consisting of the events of probability 0 or 1 only, then E{X| F} = EX;

• [(vi)] If X, Y are integrable and a, b Î IR then E{aX+bY| F} = aE{X| F}+bE{Y| F};

• [(vii)] If X and F are independent, then E{X| F} = EX.

Remark: Inequality (iv) is known as Jensen's inequality and this is how we shall refer to it.

The proof uses the following.

[ 2 If Y₁ and Y₂ are F-measurable and ò_AY₁ dP £ ò_A Y₂ dP for all A Î F, then Y₁ £ Y₂ almost surely. If ò_AY₁ dP = ò_A Y₂ dP for all A Î F, then Y₁ = Y₂.

Proof. Let A_e = {Y₁ > Y₂+e} Î F. Since ò_AeY₁ dP ³ ò_AeY₂ dP + eP(A_e), thus P(A_e) > 0 is impossible. Event {Y₁ > Y₂} is the countable union of the events A_e (with e rational); thus it has probability 0 and Y₁ £ Y₂ with probability one.

The second part follows from the first by symmetry. ^[¯]
Proof of Theorem 1.4.

(i) This is verified first for Y = I_B (the indicator function of an event B Î F). Let Y₁ = E{XY| F}, Y₂ = YE{X| F}. From the definition one can easily see that both ò_AY₁ dP and ò_A Y₂ dP are equal to ò_{A ÇB} X dP. Therefore Y₁ = Y₂ by the Lemma 1.4.

For the general case, approximate Y by simple random variables and use (vi).

(ii) This follows from Lemma 1.4: random variables Y₁ = E{X| F}, Y₂ = E{X| G} are G-measurable and for A in G both ò_AY₁ dP and ò_A Y₂ dP are equal to ò_AX dP.

(iii) Let Y₁ = E{X| NÚ F}, Y₂ = E{X| F}. We check first that

(iv) Here we need the first part of Lemma 1.4. We also need to know that each convex function g(x) can be written as the supremum of a family of affine functions f_{a, b} (x) = ax+b. Let Y₁ = E{g(X)| F}, Y₂ = f_{a, b}(E{X| F}), A Î F. By (vi) we have

(v), (vi), (vii) These proofs are left as exercises. ^[¯]

Theorem 1.4 gives geometric interpretation of the conditional expectation E{·| F} as the projection of the Banach space L_p(W, M, P) onto its closed subspace L_p(W, F, P), consisting of all p-integrable F-measurable random variables, p ³ 1. This projection is ``self adjoint'' in the sense that the adjoint operator is given by the same ``conditional expectation'' formula, although the adjoint operator acts on L_q rather than on L_p; for square integrable functions E{.| F} is just the orthogonal projection onto L₂(W, F, P). Monograph [] considers conditional expectation from this angle.

We will use the following (weak) version of the martingale³ convergence theorem.

Theorem 6 Suppose F_n is a decreasing family of s-fields, ie. F_n+1 Ì F_n for all n ³ 1. If X is integrable, then E{X| F_n}® E{X| F} in L₁-norm, where F is the intersection of all F_n.

Proof. Suppose first that X is square integrable. Subtracting m = EX if necessary, we can reduce the convergence question to the centered case EX = 0. Denote X_n = E{X| F_n}. Since F_n+1 Ì F_n, by Jensen's inequality EX_n² ³ 0 is a decreasing non-negative sequence. In particular, EX_n² converges.

Let m < n be fixed. Then E(X_n-X_m)² = EX_n²+EX_m²-2EX_nX_m. Since F_n Ì F_m, by Theorem 1.4 we have

If X is not square integrable, then for every e > 0 there is a square integrable Y such that E|X-Y| < e. By Jensen's inequality E{X| F_n} and E{Y| F_n} differ by at most e in L₁-norm; this holds uniformly in n. Since by the first part of the proof E{Y| F_n} is convergent, it satisfies the Cauchy condition in L₂ and hence in L₁. Therefore for each e > 0 we can find N such that for all n, m > N we have E{|E{X| F_n}-E{X| F_m}|} < 3e. This shows that E{X| F_n} satisfies the Cauchy condition and hence converges in L₁.

The fact that the limit is X_¥ = E{X| F} can be seen as follows. Clearly X_¥ is F_n-measurable for all n, ie. it is F-measurable. For A Î F (hence also in F_n), we have EXI_A = EX_nI_A. Since

|EX_nI_A-EX_¥ I_A| £ E|X_n-X_¥ |I_A £ E|X_n-X_¥ |® 0, therefore EX_nI_A® EX_¥ I_A. This shows that EXI_A = EX_¥ I_A and by definition, X_¥ = E{X| F}. ^[¯]

1.5  Characteristic functions

Example 1 The following gives an example of characteristic function that has finite support. Let f(t) = 1-|t| for |t < | < 1 and 0 otherwise. Then

The following properties of characteristic functions are proved in any standard probability course, see eg. [,].

Theorem 7 (i) The distribution of X is determined uniquely by its characteristic function f(t).

(ii) If E|X|^r < ¥ for some r = 0,1,¼, then f(t) is r-times differentiable, the derivative is uniformly continuous and

(iv) If X, Y are independent random variables, then f_X+Y(t) = f_X(t) f_Y(t) for all t Î IR.

For a d-dimensional random variable X = (X₁, ¼, X_d) the characteristic function f_X: IR^d® \sf CC is defined by f_X(t) = Eexp(it·X), where the dot denotes the dot (scalar) product, ie. x·y = åx_ky_k. For a pair of real valued random variables X, Y, we also write f(t, s) = f_{(X, Y)}((t, s)) and we call f(t, s) the joint characteristic function of X and Y.

The following is the multi-dimensional version of Theorem 1.5.

Theorem 8 (i) The distribution of X is determined uniquely by its characteristic function f(t).

(ii) If E||X||^r < ¥, then f(t) is r-times differentiable and

The next result seems to be less known although it is both easy to prove and to apply. We shall use it on several occasions in Chapter . The converse is also true if we assume that the integer parameter r in the proof below is even or that joint characteristic function f(t, s) is differentiable; to prove the converse, one can follow the usual proof of the inversion formula for characteristic functions, see, eg. []. Kagan, Linnik & Rao [] state explicitly several most frequently used variants of ().

Theorem 9 Suppose real valued random variables X, Y have the joint characteristic function f(t, s). Assume that E|X|^m < ¥ for some m Î IN. Let g(y) be such that

Proof. Since by assumption E|X|^m < ¥, the joint characteristic function f(t, s) = Eexp(itX+isY) can be differentiated m times with respect to t and

In order to prove (17), we need to show first that E|Y|^r < ¥, where r is the degree of the polynomial g(y). By Jensen's inequality E|g(Y)| £ E|X|^m < ¥, and since |g(y)/y^r|® const ¹ 0 as |y|® ¥, therefore there is C > 0 such that |y|^r £ C|g(y)| for all y. Hence E|Y|^r < ¥ follows.

Formula (17) is now a simple consequence of (16); indeed, for 0 £ k £ r we have EY^kexp(isY) = (-i)^kkf(0, s); this formula is obtained by differentiating k-times Eexp(isY) under the integral sign. ^[¯]

1.6  Symmetrization

Definition 2 A random variable X (also: a vector valued random variable X) is symmetric if X and -X have the same distribution.

Symmetrization techniques deal with comparison of properties of an arbitrary variable X with some symmetric variable X_sym. Symmetric variables are usually easier to deal with, and proofs of many theorems (not only characterization theorems, see eg. []) become simpler when reduced to the symmetric case.

There are two natural ways to obtain a symmetric random variable X_sym from an arbitrary random variable X. The first one is to multiply X by an independent random sign ±1; in terms of the characteristic functions this amounts to replacing the characteristic function f of X by its symmetrization ¹/₂ ( f(t)+ f(-t)). This approach has the advantage that if X is symmetric, then its symmetrization X_sym has the same distribution as X. Integrability properties are also easy to compare, because |X| = |X_sym|.

The other symmetrization, which has perhaps less obvious properties but is frequently found more useful, is defined as follows. Let X¢ be an independent copy of X. The symmetrization [X\tilde] of X is defined by [X\tilde] = X-X¢. In terms of the characteristic functions this corresponds to replacing the characteristic function f(t) of X by the characteristic function |f(t)|². This procedure is easily seen to change the distribution of X, except when X = 0.

Theorem 10 (i) If the symmetrization [X\tilde] of a random variable X has a finite moment of order p ³ 1, then E|X|^p < ¥.

(ii) If the symmetrization [X\tilde] of a random variable X has finite exponential moment Eexp(l|[X\tilde]|), then Eexpl|X| < ¥, l > 0.

(iii) If the symmetrization [X\tilde] of a random variable X satisfies Eexpl|[X\tilde]|² < ¥, then Eexpl|X|² < ¥, l > 0.

The usual approach to Theorem 1.6 uses the symmetrization inequality, which is of independent interest (see Problem ) and formula (2). Our proof requires extra assumptions, but instead is short, does not require X and X¢ to have the same distribution, and it also gives a more accurate bound (within its domain of applicability).

Proof in the case, when E|X| < ¥ and EX = 0: Let g(x) ³ 0 be a convex function, such that Eg([X\tilde]) < ¥ and let X, X¢ be the independent copies of X, so that conditional expectation E^XX¢ = EX = 0. Then Eg(X) = Eg(X-E^XX¢) = Eg(E^X{X-X¢}). Since by Jensen's inequality, see Theorem 1.4 (iv) we have Eg(E^X{X-X¢}) £ Eg(X-X¢), therefore Eg(X) £ Eg(X-X¢) = Eg([X\tilde]) < ¥. To end the proof, consider three convex functions g(x) = |x|^p, g(x) = exp(lx) and g(x) = exp(lx²).

1.7  Uniform integrability

Uniform integrability is often used in conjunction with weak convergence to verify the convergence of moments. Namely, if X_n is uniformly integrable and converges in distribution to Y, then Y is integrable and

[ 3 If X₁,X₂,... are centered i. i. d. random variables with finite second moments and S_n = å_{j = 1}ⁿX_j then {¹/_nS_n²}_{n ³ 1} is uniformly integrable.

The following lemma is a special case of the celebrated Khinchin inequality.

[ 3 If e_j are ±1 valued symmetric independent r. v., then for all real numbers a_j

Proof. By independence and symmetry we have

[ 4 If X_k are i. i. d. centered with fourth moments, then there is a constant C < ¥ such that

Proof. As in the proof of Theorem 1.6 we can estimate the fourth moments of a centered r. v. by the fourth moment of its symmetrization, ES_n⁴ £ E[S\tilde]_n⁴.

Let e_j be independent of [X\tilde]_k's as in Lemma 1.7. Then in distribution [S\tilde]_n @ å_{j = 1}ⁿe_j[X\tilde]_j. Therefore, integrating with respect to the distribution of e_j first, from (19) we get

[ 5 If U,V ³ 0 then

Proof. By (2) applied to f(x) = x² I_{x > 2t} we have

Let e > 0 and choose M > 0 such that ò_{{|X| > M}}|X| dP < e. Split X_k = X_k^¢+X_k^¢¢, where X_k^¢ = X_kI_{{|Xk| £ M}}-E{X_kI_{{|Xk| £ M}}} and let S¢, S¢¢ denote the corresponding sums.

Notice that for any U ³ 0 we have UI_{{|U| > m}} £ U²/m. Therefore ¹/_nò_{|Sn¢| > t Ön}(S_n¢)² dP £ t^-2n^-2E(S_n¢)⁴, which by Lemma 1.7 gives

Now we use orthogonality to estimate the second term:

limsup_t®¥sup_n¹/_nò_{{|Sn| > 2tÖn}}S_n² dP £ e. Since e > 0 is arbitrary, this ends the proof. ^[¯]

1.8  The Mellin transform

Definition 3 ⁵ The Mellin transform of a random variable X ³ 0 is defined for all complex s such that EX^{Âs -1} < ¥ by the formula M(s) = EX^s-1.

The definition is consistent with the usual definition of the Mellin transform of an integrable function: if X has a probability density function f(x), then the Mellin transform of X is given by M(s) = ò₀^¥ x^s-1f(x) dx.

Theorem 11 ⁶ If X ³ 0 is a random variable such that EX^a-1 < ¥ for some a ³ 1, then the Mellin transform M(s) = EX^s-1, considered for s Î \sf CC such that Âs = a, determines the distribution of X uniquely.

Proof. The easiest case is when a = 1 and X > 0. Then M (s) is just the characteristic function of log(X); thus the distribution of log(X), and hence the distribution of X, is determined uniquely.

In general consider finite non-negative measure m defined on (IR₊, B) by

Theorem 12 If X ³ 0 and EX^a < ¥ for some a > 0, then the Mellin transform of X is analytic in the strip 1 < Âs < 1+a.

Proof. Since for every s with 0 < Âs < a the modulus of the function w® X^slog(X) is bounded by an integrable function C₁+C₂|X|^a, therefore EX^s can be differentiated with respect to s under the expectation sign at each point s, 0 < Âs < a. ^[¯]

1.9  Problems

Problem 1 [[]] Use Fubini's theorem to show that if XY, X, Y are integrable, then

Problem 2 Let X ³ 0 be a random variable and suppose that for every 0 < q < 1 there is T = T(q) such that

Problem 3 Show that if X ³ 0 is a random variable such that

Problem 4 Show that if Eexp(lX²) = C < ¥ for some a > 0, then

Problem 5 Show that (11) implies E{|X|^|X|} < ¥.

Problem 6 Prove part (v) of Theorem 1.4.

Problem 7 Prove part (vi) of Theorem 1.4.

Problem 8 Prove part (vii) of Theorem 1.4.

Problem 9 Prove the following conditional version of Chebyshev's inequality: if F is a s-field, and E|X| < ¥, then

Problem 10 Show that if (X, Y) is uniformly distributed on a circle centered at (0, 0), then for every a, b there is a non-random constant C = C(a, b) such that E{X|aX+bY} = C(a,b)(aX+bY).

Problem 11 Show that if (U,V,X) are such that in distribution (U,X) @ (V,X) then E{U|X} = E{V|X} almost surely.

Problem 12 Show that if X, Y are integrable non-degenerate random variables, such that

Problem 13 Suppose that X, Y are square-integrable random variables such that

Problem 14 Show that if X, Y are integrable such that E{X|Y} = Y and E{Y|X} = X, then X = Y a. s.

Problem 15 Prove that if X ³ 0, then function f(t): = EX^it, where t Î IR, determines the distribution of X uniquely.

Problem 16 Prove that function f(t): = Emax{X, t} determines uniquely the distribution of an integrable random variable X in each of the following cases:

• [(a)] If X is discrete.

• [(b)] If X has continuous density.

Problem 17 Prove that, if E|X| < ¥, then function f(t): = E|X-t| determines uniquely the distribution of X.

Problem 18 Let p > 2 be fixed. Show that exp(-|t|^p) is not a characteristic function.

Problem 19 Let Q(t,s) = logf(t,s), where f(t,s) is the joint characteristic function of square-integrable r. v. X,Y.

• [(i)] Show that E{X|Y} = rY implies

  ¶ ¶t Q(t,s) ê ê ê t = 0  = r d ds Q(0,s).

• [(ii)] Show that E{X²|Y} = a+bY+cY² implies

  ¶2 ¶t2 Q(t,s) ê ê ê t = 0  + æ ç è ¶ ¶t Q(t,s) ö ÷ ø 2   ê ê ê t = 0

  = -a +ib d ds Q(0,s)+c d2 d s2 Q(0,s)+c æ ç è d d s Q(0,s) ö ÷ ø 2   .

Problem 20 [see eg. []] Suppose a Î IR is the median of X.

• [(i)] Show that the following symmetrization inequality

  P(|X| ³ t) £ 2P(| ~ X   | ³ t -|a|)

  holds for all t > |a|.

• [(ii)] Use this inequality to prove Theorem 1.6 in the general case.

Problem 21 Suppose (X_n,Y_n) converge to (X,Y) in distribution and {X_n}, {Y_n} are uniformly integrable. If E(X_n|Y_n) = rY_n for all n, show that E(X|Y) = rY.

Problem 22 Prove (18).

Chapter 2
Normal distributions

2.1  Univariate normal distributions

In multivariate case it is more convenient to use characteristic functions directly. Besides, characteristic functions are our main technical tool and it doesn't hurt to start using them as soon as possible. We shall therefore begin with the following definition.

Definition 4 A real valued random variable X has the normal N(m, s) distribution if its characteristic function has the form

From Theorem 1.5 it is easily to check by direct differentiation that m = EX and s² = Var(X). Using (15) it is easy to see that every univariate normal X can be written as

The following properties of standard normal distribution N(0,1) are self-evident:

1. The characteristic function e^{-[(t2)/ 2]} has analytic extension e^{-[(z2)/ 2]} to all complex z Î \sf CC. Moreover, e^{-[(z2)/ 2]} ¹ 0.
2. Standard normal random variable Z has finite exponential moments Eexp(l|Z|) < ¥ for all l; moreover, Eexp(lZ²) < ¥ for all l < ¹/₂ (compare Problem 1.9).

For future reference we state the following simple but useful observation. Computing EX^k for k = 0, 1, 2 from Theorem 1.5 we immediately get.

[ 4 A characteristic function which can be expressed in the form f(t) = exp(at²+bt+c) for some complex constants a, b, c, corresponds to the normal random variable, ie. a Î IR and a < 0, b Î i IR is imaginary and c = 0.

2.2  Multivariate normal distributions

We follow the usual linear algebra notation. Vectors are denoted by small bold letters x, v, t, matrices by capital bold initial letters A, B, C and vector-valued random variables by capital boldface X, Y, Z; by the dot we denote the usual dot product in IR^d, ie. x·y: = å_{j = 1}^d x_jy_j; ||x|| = (x·x)^1/2 denotes the usual Euclidean norm. For typographical convenience we sometimes write (a₁,¼,a_k) for the vector

[

a1

:

ak

]. By A^T we denote the transpose of a matrix A.

Below we shall also consider another scalar product á·,·ñ associated with the normal distribution; the corresponding semi-norm will be denoted by the triple bar |||·|||.

Definition 5 An IR^d-valued random variable Z is multivariate normal, or Gaussian (we shall use both terms interchangeably; the second term will be preferred in abstract situations) if for every t Î IR^d the real valued random variable t·Z is normal.

Clearly the distribution of univariate t·Z is determined uniquely by its mean m = m_t and its standard deviation s = s_t. It is easy to see that m_t = t·m, where m = EZ. Indeed, by linearity of the expected value m_t = Et·Z = t·EZ. Evaluating the characteristic function f(s) of the real-valued random variable t·Z at s = 1 we see that the characteristic function of Z can be written as

The following observations are easy to check.

• B(·, ·) is symmetric, ie. B(x, y) = B(y, x) for all x, y;

• B(·, ·) is a bilinear function, ie. B(·, y) is linear for every fixed y and B(x, ·) is linear for very fixed x;

• B(·, ·) is positive definite, ie. B(x, x) ³ 0 for all x.

We shall need the following well known linear algebra fact (the proofs are explained below; explicit reference is, eg. []).

[ 6 Each bilinear form B has the dot product representation

Indeed, expand x and y with respect to the standard orthogonal basis e ₁,¼, e _d. By bilinearity we have B(x, y) = å_i,jx_iy_j B( e _i, e _j), which gives the dot product representation with c_{i, j} = B( e _i, e _j). Clearly, for symmetric B(·, ·) we get c_{i, j} = c_{j, i}; hence C is symmetric.

[ 7 If in addition B(·, ·) is positive definite then

for a d×d matrix A. Moreover, A can be chosen to be symmetric.

The easiest way to see the last fact is to diagonalize C (this is always possible, as C is symmetric). The eigenvalues of C are real and, since B(·, ·) is positive definite, they are non-negative. If L denotes a (diagonal) matrix (consisting of eigenvalues of C) in the diagonal representation C = ULU^T and D is the diagonal matrix formed by the square roots of the eigenvalues, then A = UDU^T. Moreover, this construction gives symmetric A = A^T. In general, there is no unique choice of A and we shall sometimes find it more convenient to use non-symmetric A, see Example below.

The linear algebra results imply that the characteristic function corresponding to a normal distribution on IR^d can be written in the form

Theorem 13 The characteristic function corresponding to a normal random variable Z = (Z₁, ¼, Z_d) is given by (25), where m = EZ and C = [c_{i, j}], c_{i, j} = Cov(Z_i, Z_j), is the covariance matrix.

From (24) and (25) we get also

From the above discussion, we have the following multivariate generalization of (23).

Theorem 14 Each d-dimensional normal random variable Z has the same distribution as m+A[(g)\vec], where m Î IR^d is deterministic, A is a (symmetric) d×d matrix and [(g)\vec] = (g₁, ¼, g_d) is a random vector such that the components g₁, ¼, g_d are independent N(0, 1) random variables.

Proof. Clearly, Eexp( it·(m+A[(g)\vec])) = exp(it·m) Eexp( it·(A[(g)\vec])). Since the characteristic function of [(g)\vec] is Eexp( ix·[(g)\vec]) = exp-¹/₂ || x|| ² and t·(A[(g)\vec]) = (A^Tt)·[(g)\vec], therefore we get

Eexp( it·(m+A[(g)\vec])) = expit·mexp -¹/₂ || A^Tt|| ², which is another form of (26). ^[¯]

Theorem 2.2 can be actually interpreted as the almost sure representation. However, if A is not of full rank, the number of independent N(0,1) r. v. can be reduced. In addition, the representation Z @ m+A[(g)\vec] from Theorem 2.2 is not unique if the symmetry condition is dropped. Theorem gives the same representation with non-symmetric A = [e₁,¼,e_k]. The argument given below has more geometric flavor. Infinite dimensional generalizations are also known, see () and the comment preceding Lemma .

Theorem 15 Each d-dimensional normal random variable Z can be written as

Proof. Without loss of generality we may assume EZ = 0 and establish the representation with m = 0.

Let IH denote the linear span of the columns of A in IR^d, where A is the matrix from (26). From Theorem 2.2 it follows that with probability one Z Î IH. Consider now IH as a Hilbert space with a scalar product áx, yñ, given by áx, yñ = (Ax)·(Ay). Since the null space of A and the column space of A have only zero vector in common, this scalar product is non-degenerate, ie. áx, xñ ¹ 0 for IH ' x ¹ 0.

Let e ₁, e ₂, ¼, e _k be the orthonormal (with respect to á·,·ñ) basis of IH, where k = dimIH. By Theorem 2.2 Z is IH-valued. Therefore with probability one we can write Z = å_{j = 1}^k g_j e _j, where g_j = áe _j,Zñ are random coefficients in the orthogonal expansion. It remains to verify that g₁, ¼, g_k are i. i. d. normal N(0, 1) r. v. With this in mind, we use (26) to compute their joint characteristic function:

The next theorem lists two important properties of the normal distribution that can be easily verified by writing the joint characteristic function. The second property is a consequence of the polarization identity

Theorem 16 If X, Y are independent with the same centered normal distribution, then

a) [(X+Y)/( Ö2)] has the same distribution as X;

b) X+Y and X-Y are independent.

Now we consider the multivariate normal density. The density of [(g)\vec] in Theorem 2.2 is the product of the one-dimensional standard normal densities, ie.

Theorem 17 If Z is centered normal with the nonsingular covariance matrix C, then the density of Z is given by

In the nonsingular case this immediately implies strong integrability.

Theorem 18 If Z is normal, then there is e > 0 such that

Remark: Theorem 2.2 holds true also in the singular case and for Gaussian random variables with values in infinite dimensional spaces; for the proof based on Theorem 2.2, see Theorem below.

The Hilbert space IH introduced in the proof of Theorem 2.2 is called the Reproducing Kernel Hilbert Space (RKHS) of a normal distribution, cf. [,]. It can be defined also in more general settings. Suppose we want to consider jointly two independent normal r. v. X and Y, taking values in IR^d₁ and IR^d₂ respectively, with corresponding reproducing kernel Hilbert spaces IH₁, IH₂ and the corresponding dot products á·,·ñ₁ and á·,·ñ₂. Then the IR^d₁+d₂-valued random variable (X, Y) has the orthogonal sum IH₁ÅIH₂ as the Reproducing Kernel Hilbert Space.

This method shows further geometric aspects of jointly normal random variables. Suppose an IR^d₁+d₂-valued random variable (X, Y) is (jointly) normal and has IH as the reproducing kernel Hilbert space (with the scalar product á·,·ñ). Recall that

IH = {A[

x

y

]: [

x

y

] Î IR^d₁+d₂}. Let IH_Y be the subspace of IH spanned by the vectors

{[

0

y

]: y Î IR^d₂}; similarly let IH_X be the subspace of IH spanned by the vectors

[

x

0

]. Let P be (the matrix of) the linear transformation IH_X® IH_Y obtained from the á·,·ñ-orthogonal projection IH® IH_X by narrowing its domain to IH_X. Denote Q = P^T; Q represents the orthogonal projection in the dual norm defined in Section below.

Theorem 19 If (X, Y) has jointly normal distribution on IR^d₁+d₂, then random vectors Y-QY and X are stochastically independent.

Proof. The joint characteristic function of X-QY and Y factors as follows:

[

0

s

] and

[

t

-Pt

] are orthogonal with respect to scalar product á·,·ñ. ^[¯]
In particular, since E{X| Y} = E{X-QY| Y}+QY, we get

[ 7 If both X and Y have mean zero, then

For general multivariate normal random variables X and Y applying the above to centered normal random variables X-m_X and Y-m_Y respectively, we get

C = [

CX	R
RT	CY

]. An alternative proof of (30) (and of Corollary 2.2) is to use the converse to Theorem 1.5.

Equality (30) is usually referred to as linearity of regression. For the bivariate normal distribution it takes the form E{X|Y} = a+bY and it can be established by direct integration; for more than two variables computations become more difficult and the characteristic functions are quite handy.

[ 8 Suppose (X, Y) has a (joint) normal distribution on IR^d₁+d₂ and IH_X, IH_Y are á,·,·ñ-orthogonal, ie. every component of X is uncorrelated with all components of Y. Then X, Y are independent.

Indeed, in this case Q is the zero matrix; the conclusion follows from Theorem 2.2.

Example 2 In this example we consider a pair of (jointly) normal random variables X₁, X₂. For simplicity of notation we suppose EX₁ = 0, EX₂ = 0. Let Var(X₁) = s₁², Var(X₂) = s₂ ² and denote corr(X₁, X₂) = r. Then

C = [

s12	r
r	s22

] and the joint characteristic function is

Returning back to our original random variables X₁, X₂, we have X₁ = g₁ s₁cosq+ g₂ s₁sinq and X₂ = g₁ s₂sinq+ g₂ s₂cosq; this representation holds true also in the degenerate case.

To illustrate previous theorems, notice that Corollary 2.2 in the bivariate case follows immediately from (31). Theorem 2.2 says in this case that Y₁-rY₂ and Y₂ are independent; this can also be easily checked either by using density (31) directly, or (easier) by verifying that Y₁-rY₂ and Y₂ are uncorrelated.

Example 3 In this example we analyze a discrete time Gaussian random walk {X_k}_{0 £ k £ T}. Let x₁,x₂,¼ be i. i. d. N(0,1). We are interested in explicit formulas for the characteristic function and for the density of the IR^T-valued random variable X = (X₁, X₂, ¼, X_T), where

2.3  Analytic characteristic functions

Definition 6 We shall say that a characteristic function f(t) is analytic if it can be extended from the real line IR to the function analytic in a domain in complex plane \sf CC.

Because of uniqueness we shall use the same symbol f to denote both.

Clearly, normal distribution has analytic characteristic function. Example 1.5 presents a non-analytic characteristic function.

We begin with the probabilistic (moment) condition for the existence of the analytic extension.

Theorem 20 If a random variable X has finite exponential moment Eexp(a|X|) < ¥, where a > 0, then its characteristic function f(s) is analytic in the strip -a < Ás < a.

Proof. The analytic extension is given explicitly: f(s) = Eexp(isX). It remains only to check that f(s) is differentiable in the strip -a < Ás < a. This follows either by differentiation with respect to s under the expectation sign (the latter is allowed, since E{|X|exp(|sX|)} < ¥, provided -a < Ás < a), or by writing directly the series expansion: f(s) = å_{n = 0}^¥ iⁿEXⁿ sⁿ/n! (the last equality follows by switching the order of integration and summation, ie. by Fubini's theorem). The series is easily seen to be absolutely convergent for all -a £ Ás £ a. ^[¯]

[ 9 If X is such that Eexp(a|X|) < ¥ for every real a > 0, then its characteristic function f(s) is analytic in \sf CC.

The next result says that normal distribution is determined uniquely by its moments. For more information on the moment problem, the reader is referred to the beautiful book by N. I. Akhiezer [].

[ 10 If X is a random variable with finite moments of all orders and such that EX^k = EZ^k, k = 1, 2,¼, where Z is normal, then X is normal.

Proof. By the Taylor expansion

[ 11 Let f(t) be a characteristic function, and suppose there is s² > 0 and a sequence {t_k} convergent to 0 such that f(t_k) = exp(-s²t_k²) and t_k ¹ 0 for all k. Then f(t) = exp(-s²t²) for every t Î IR.

Proof. The idea of the proof is simply to calculate all the derivatives at 0 of f(t) along the sequence {t_k}. Since the derivatives determine moments uniquely, by Corollary 2.3 we shall conclude that f(t) = exp(- s² t²). The only nuisance is to establish that all the moments of the distribution are finite. This fact is established by modifying the usual proof of Theorem 1.5(iii). Let D_t² be a symmetric second order difference operator, ie.

Claim 1 If f(t) is the characteristic function of a random variable X, t(k)® 0 is a given sequence such that t(k) ¹ 0 for all k and

The proof of the claim rests on the formula which can be verified by elementary calculations:

Theorem 21 If the characteristic function f(t) of a random variable X has the analytic extension in a neighborhood of 0 in \sf CC, and the extension is such that the Taylor expansion series at 0 has convergence radius R £ ¥, then Eexp(a|X|) < ¥ for all 0 £ a < R.

Proof. By assumption, f(s) has derivatives of all orders. Thus the moments of all orders are finite and

[ 12 If a characteristic function f(t) can be extended analytically to the circle |s| < a, then it has analytic extension f(s) = Eexp( isX) to the strip -a < Ás < a.

2.4  Hermite expansions

A normal N(0,1) r. v. Z defines a dot product áf, g ñ = Ef(Z)g(Z), provided that f(Z) and g(Z) are square integrable functions on W. In particular, the dot product is well defined for polynomials. One can apply the usual Gram-Schmidt orthogonalization algorithm to functions 1, Z, Z², ¼. This produces orthogonal polynomials in variable Z known as Hermite polynomials. Those play important role and can be equivalently defined by

Hermite polynomials actually form an orthogonal basis of L₂(Z). In particular, every function f such that f(Z) is square integrable can be expanded as f(x) = å_{n = 1}^¥ f_k H_k(x), where f_k Î IR are Fourier coefficients of f(·); the convergence is in L₂(Z), ie. in weighted L₂ norm on the real line, L₂(IR, e^-x2/2dx).

The following is the classical Mehler's formula.

Theorem 22 For a bivariate normal r. v. X,Y with EX = EY = 0, EX² = EY² = 1, EXY = r, the joint density q(x,y) of X,Y is given by

Proof. By Fourier's inversion formula we have

2.5  Cramer and Marcinkiewicz theorems

[ 8 If X is a random variable such that Eexp(lX²) < ¥ for some l > 0, and the analytic extension f(z) of the characteristic function of X satisfies f(z) ¹ 0 for all z Î \sf CC, then X is normal.

Proof. By the assumption, f(z) = logf(z) is well defined and analytic for all z Î \sf CC. Furthermore if z = x+ iy is the decomposition of z Î \sf CC into its real and imaginary parts, then Âf(z) = log| f(z)| £ log(Eexp|yX|). Notice that Eexp(tX) £ Cexp([(t²)/( 2l)]) for all real t, see Problem 1.9. Indeed, since lX²+t²/l ³ 2tX, therefore Eexp(tX) £ Eexp(lX²+t²/a)/2 = Cexp([(t²)/( 2l)]). Those two facts together imply Âf(z) £ const+[(y²)/ 2a]. Therefore a variant of the Liouville theorem [] implies that f(z) is a quadratic polynomial in variable z, ie. f(z) = A+Bz+Cz². It is easy to see that the coefficients are A = 0, B = i E{X}, C = -Var(X)/2, compare Proposition 2.1. ^[¯]
From Lemma 2.5 we obtain quickly the following important theorem, due to H. Cramer [].

Theorem 23 If X₁ and X₂ are independent random variables such that X₁+X₂ has a normal distribution, then each of the variables X₁, X₂ is normal.

Theorem 2.5 is celebrated Cramer's decomposition theorem; for extensions, see []. Cramer's theorem complements nicely the Central Limit Theorem in the following sense. While the Central Limit Theorem asserts that the distribution of the sum of i. i. d. random variables with finite variances is close to normal, Cramer's theorem says that it cannot be exactly normal, except when we start with a normal sequence. This resembles propagation of chaos phenomenon, where one proves a dynamical system approaches chaotic behavior, but it never reaches it except from initially chaotic configurations. We shall use Theorem 2.5 as a technical tool.

Proof of Theorem 2.5. Without loss of generality we may assume EX₁ = EX₂ = 0. The proof of Theorem 1.6 (iii) implies that Eexp(aX_j²) < ¥, j = 1, 2. Therefore, by Theorem 2.3, the corresponding characteristic functions f₁ (·), f₂ (·) are analytic. By the uniqueness of the analytic extension, f₁(s)f₂(s) = exp(-s²/2) for all s Î \sf CC. Thus f_j(z) ¹ 0 for all z Î \sf CC, j = 1, 2, and by Lemma 2.5 both characteristic functions correspond to normal distributions. ^[¯]
The next theorem is useful in recognizing the normal distribution from what at first sight seems to be incomplete information about a characteristic function. The result and the proof come from Marcinkiewicz [], cf. [].

Theorem 24 Let Q(t) be a polynomial, and suppose that a characteristic function f has the representation f(t) = expQ(t) for all t close enough to 0. Then Q is of degree at most 2 and f corresponds to a normal distribution.

Proof. First note that formula f(s) = expQ(s), s Î \sf CC, defines the analytic extension of f. Thus, by Corollary 2.3, f(s) = Eexp(isX), s Î \sf CC. By Theorem 2.5, it suffices to show that f(s) f(-s) corresponds to the normal distribution. Clearly f(s) f(-s) also has the form exp( P(t)), where P(s) is a polynomial that has only even terms, ie. P(s) = å_{k = 0}ⁿ a_ks^2k. Since f(s)f(-s) = | f(s)|² is a real number for all s, the coefficients a₁, ¼, a_n of polynomial P (·) are real. Moreover, the n-th coefficient satisfies a_n = -g² < 0, as the inequality | f(t)| £ 1 holds for arbitrarily large real t. Therefore, taking z = N exp( ip/(2n)), we obtain

2.6  Large deviations

Here we shall present the logarithmic term in the asymptotic expansion for P(X Î nA) as n® ¥. This is the so called large deviation estimate; it becomes more accurate for less likely events. The main feature is that it has relatively simple form and applies to all events. Higher order expansions are more accurate but work for fairly regular sets A Ì IR^d only.

Let us first define the conjugate ``norm'' to the RKHS seminorm |||·||| defined by (29).

The conjugate norm has all the properties of the norm except that it can attain value ¥. To see this, and also to have a more explicit expression, decompose I\negthinspace R^d into the orthogonal sum of the null space of A and the range of A: IR^d = N(A)ÅÂ(A); here A is the symmetric matrix from (26). Since A:IR^d® Â(A) is onto, there is a right-inverse A^-1:Â(A)® Â(A) Ì I\negthinspace R^d.

For y Î Â(A) we have

For y\not Î Â(A) we write y = y _N+y_Â, where 0 ¹ y _N Î N(A). Then we have sup_{||Ax|| = 1}x·y ³ sup_{x Î N(A)}x·y _N = ¥. Since C = A×A, we get

In this notation, the multivariate normal density is