A variational formula on the Cramér function of series of independent random variables

In (Zajkowski, Positivity 19:529–537, 2015) it has been proved some variational formula on the Legendre–Fenchel transform of the cumulant generating function (the Cramér function) of Rademacher series with coefficients in the space ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^1$$\end{document}. In this paper we show a generalization of this formula to series of a larger class of any independent random variables with coefficients that belong to the space ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^2$$\end{document}.


Introduction
The Legendre-Fenchel transforms of cumulant generating functions of given random variables are at the core of the large deviations theory (see e.g. [3,4]). The Cramér function gives the rate of the exponentially decay of tails of distributions for the empirical means of sequences of i.i.d. random variables. It provides a nice connection between convex analysis and statistics.
Donsker and Varadhan in [5] proved that the Legendre-Fenchel transform ψ * X of the cumulant generating function ψ X (s) = ln Ee s X of a random variable X satisfies the following variational principle where D(m μ X ) = ln dm dμ X dm is the relative entropy of a probability distribution m with respect to the distribution μ X of X .
The aim of this paper is to prove some variational formula for the Cramér functions of series of independent random variables that depends on coefficients and Cramér functions of summands of a given series; see Theorem 2.3.
For series t i g i of independent standard normal r.v.s, where t 2 i < ∞ and g i ∈ N (0, 1), it is known their tail estimation of the form ; see for instance [10]. The function α 2 2 t 2 i is the Cramér function of the random series To realize our purposes we will need the general notion of the Legendre-Fenchel transform in topological spaces (see [6] or [1]). Let V be a real locally convex Hausdorff space and V * its dual space. By ·, · we denote the canonical pairing between V and V * .
is called the Legendre-Fenchel transform (convex conjugate) of f and a function f * * : is called the convex biconjugate of f . The functions f * and f * * are convex and lower semicontinuous in the weak* and weak topology on V * and V , respectively. Moreover, the biconjugate theorem states that the function f : V → R ∪ {∞} not identically equal to +∞ is convex and lower semicontinuous if and only if f = f * * .
Let us mention additional properties of the convex conjugates; see 4.3 Examples in [6]. Let V be a normed space. We denote by · the norm of V and by · * the norm of V * . For conjugate exponents p, q ∈ (1, ∞) ( 1 p + 1 q = 1), a function 1 q u * q * is the convex conjugate of 1 p u p .
Remark 1.1 Let us emphasize that in Hilbert spaces a function 1 2 u 2 one can treat as the function invariant with respect to the Legendre-Fenchel transform.
Let us list two properties and the notion of the infimal convolution. The convexconjugation is order-reversing: Let functions f 1 , . . . , f n are convex, lower semicontinuous and not identically equal . . , f n−1 are continuous. Then the convex conjugate of their sum is given by the so called infimal (in this case even minimal) convolution, i.e.
(see e.g. [7, Th. 1]). It will turn out (see Remark 2.8) that the variational formula for the Cramér function of series of independent random variables is an example of an application of a generalization of the infimal convolution to the infinite case of summands. Often generalizations of formulas from finite numbers parameters (variables) to the case of infinite ones are not obvious. Other examples of generalizations of the convex conjugates of the logarithm of series of analytic functions, with applications to investigations of the convex conjugates of the spectral radius of the functions of weighted composition operators, one can find in [8,12].

Main theorem
The cumulant generating function ψ X (s) = ln Ee s X of any random variable X is convex and lower semicontinuous on R (analytic on intD(ψ X )). It maps R into R∪{∞} and takes value zero at zero but it is possible that ψ X (s) = ∞ when s = 0. We will assume that it is finite on some neighborhood of zero, i.e. X satisfies condition: ∃ λ>0 s.t. Ee λ|X | < ∞. Let us emphasize that if E X = 0 then ψ X ≥ 0 but the Cramér function ψ * X is always nonnegative and attains 0 at the value E X. Let I ⊂ N and (X i ) i∈I be a sequence of independent r.v.s. For t = (t i ) i∈I ∈ 2 (I ) ≡ 2 consider X t = i∈I t i X i (convergence in L 2 and almost surely). The cumulant generating function of X t we will denote by ψ t , i.e. ψ t = ψ X t . Notice that for fixed s we can consider ψ t (s) as a functional of the variable t in 2 . We will denote it by ψ s . Let us emphasize that ψ s (t) = ψ t (s).
Before proving our main theorem, we show forms of the cumulant generating function and the Legendre-Fenchel transform of series of independent random variables.

Proposition 2.1
Let (X i ) i∈I be a sequence of zero-mean independent random variables with common bounded second moments. Let for each i ∈ I the cumulant generating function ψ i (s) := ψ X i (s) = ln Ee s X i is finite on some neighborhood of zero. Then for each t = (t i ) i∈I ∈ 2 the cumulant generating function of a random series X t = i∈I t i X i is given by the following equality Proof Because (X i ) are independent, centered and have common bounded second moments then for every t ∈ 2 the series X t = i∈I t i X i converges in L 2 and a.s.. Let us emphasize that the convergence of series X t in L 2 is equivalent to the convergence of sequences t in 2 . Fixing s we can consider the cumulant generating function ψ t (s) as a functional of the variable t in 2 . We will denote it by ψ s (t), i.e. ψ s (t) = ψ t (s). We show that for every s ∈ R the functional ψ s is convex and lower semicontinuous on 2 .
Convexity one may check by using the Hölder inequality. Let t, u ∈ 2 and λ ∈ (0, 1) then By the Hölder inequality for exponents 1/λ and 1/( Lower semicontinuity follows from Fatou's lemma. Let t n → t 0 in 2 . Note that X t n converges a.s. to X t 0 . Then It means that ψ s is lower semicontinuous on 2 .
Let 0 denote the space of sequences with finite supports. Observe that 0 is a dense subset of 2 . For t ∈ 0 we have For t ∈ 2 consider a series i∈I ψ i (st i ). Since E X i = 0, ψ i ≥ 0, it follows that i∈I ψ i (st i ) is convergent or divergent to plus infinity. Since ψ i are convex, this series defines a convex function on the whole 2 . Let t n → t 0 in 2 . Hence for every i ∈ I t n i → t 0 i . By superadditivity of the limit inferior and, next, by lower semicontinuity of each ψ i , we get Notice that both functions: ψ s (t) and the series i∈I ψ i (st i ) are convex and lower semicontinuous on 2 and moreover coincide on 0 (a dense subset of 2 ). It follows that these functions are equal on whole 2 , i.e.
for every t in 2 .
Let us observe that for s = 0 we have ψ 0 ≡ 0 and its convex conjugate (ψ 0 ) * (a) = 0 for a = 0 and ∞ otherwise. From now on we assume that s = 0. A form of (ψ s ) * for s = 0 is described in the following: Define now a functional i∈I ψ * i ( a i s ) on whole space 2 . Since ψ * i 's are convex and lower semicontinuous, this functional is convex and, similarly as in the case of i∈I ψ i , one can show that it is also lower semicontinuous on 2 . Because this functional coincides with (ψ s ) * on the dense subspace 0 then both functionals are equal on 2 .
Let us emphasize that the functions (ψ s ) * are nonnegative and lower semicontinuous. In large deviation theory such functions are called rate functions (good rate functions when sublevel sets are not only closed but also compact). In the main theorem below we show that the contraction principle applied to the function (ψ 1 ) * by using a functional t, · over 2 gives the Cramér function of X t .

Theorem 2.3
Let a sequence of r.v.'s (X i ) i∈I satisfies the assumptions of Proposition 2.1. Then for every t = (t i ) i∈I ∈ 2 the Cramér function ψ * X t = ψ * t of a random series X t = i∈I t i X i is given by the following variational formula Proof The functional ψ s is convex and lower semicontinuous on 2 . By virtue of the biconjugate theorem we have and we can rewrite the above as follows Let us return to the function ψ t which is convex and lower semicontinuous on R. By the biconjugate theorem we have Let us recall that ψ t (s) = ψ s (t). If we split the supremum of (4) into two parts: over R and hyperplanes {b ∈ 2 : b, t = constant} then we get Let ϕ t (α) denote the function inf b∈ 2 : t,b =α (ψ 1 ) * (b). The functional (ψ 1 ) * is convex on 2 . Convexity is preserved under contraction by linear transformation (see [4, Th. III.32]). It suffices to state that ϕ t and ψ * t coincide on intD(ψ * t ) (both ones take ∞ on a complement of clD(ψ * t )) that is Remark 2. 4 We cannot prove in general that ϕ t is lower semicontinuous. Sometimes it is obvious when for instance the effective domain of ψ * t is open subset of R (or even whole R; see Examples 2.5 and 2.6). Under an assumption that (ψ 1 ) * is a good rate function with respect to weak* topology we can state lower-semicontinuity of ϕ t (see Example 2.7). The function s 2 2 is invariant with respect to Legendre transform (see Remark 1.1) that is the Cramér function of g is given by By virtue of Proposition 2.1 the cumulant generating function of the series X t = i∈I t i g i , where g i are independent and standard normal distributed, equals By the scaling property (3) we obtain the evident form of the Cramér function On the other hand by Theorem 2.3 we get Using the Lagrange multipliers technique one can check that this infimum is attained at a sequence b = ( αt i t 2 ) i∈I .

Example 2.6
Let X be r.v. with the Laplace density 1 2 e −|x| . Its moment generating function Ee s X = 1 1−s 2 for |s| < 1 and ∞ otherwise. Let us observe that where g is standard normal distributed. Thus ψ X ≥ ψ g and, since the Legendre-Fenchel transform is order-reversing, we have Moreover by using the classical Legendre transform we can calculate an evident form of ψ * X and get Consider a sequence (X i ) i∈I of independent r.v.s with the same Laplace distribution. By (5) we have It means that (ψ 1 ) * takes finite values on the whole space 2 and for every α ∈ R there is a finite infimum: that is the finite value of ψ * t at α.
In the paper [11] one can find an example of the variational formula for the Cramér function of series of weighted symmetric Bernoulli random variables but with coefficients belonging to the space 1 . In the context of our Theorem 2.3, we recall the main result of this paper but now with coefficients in the larger space 2 .
Example 2.7 If X is a symmetric Bernoulli r.v., i.e. Pr(X = ±1) = 1 2 , then Ee s X = cosh s. By power series expansions one has cosh s ≤ exp( s 2 2 ). In this example, conversely as in previous one, we get that ψ X (s) ≤ ψ g (s) = s 2 2 and One can check that for |α| ≤ 1 and ∞ otherwise; we take 0 ln 0 = 0. Note that ψ * X (±1) = ln 2. For a sequence of independent Bernoulli r.v.s, by the above inequality, we have Since (ψ 1 ) * is lower semicontinuous in the weak* topology, sublevel sets {b ∈ 2 : (ψ 1 ) * (b) ≤ c} are weak* closed; for c < 0 are empty sets. By the above, for each c ≥ 0 the sublevel set of (ψ 1 ) * is contained in a closed ball B(0; By virtue of the Banach-Alaoglo theorem (see [9,Th. 3.15]) balls B(0; √ 2c) are weak* compact (are polar sets of balls B(0; 1/ √ 2c)). It follows that the sublevel sets {b ∈ 2 : (ψ 1 ) * (b) ≤ c} are weak* compact as closed subsets of compact sets. It means that in this topology (ψ 1 ) * is a good rate function and by the contraction principle (see [3,Th. 4.2.1]) ϕ t is also good a rate function on R, in particular, it is lower semicontinuous.
Let us emphasize that if we know that (ψ 1 ) * is a good rate function then we can prove lower-semicontinuity of ϕ t .

Remark 2.8
Let assume now that I is a finite set and Y i = t i X i then X t = i∈I Y i . Note that ψ t (s) = i∈I ψ Y i (s).
The functions ψ Y i , i ∈ I , are convex, lower semicontinuous, not identically equals +∞ and continuous at 0 that is ones satisfy the assumption of Theorem 1 [7]. The convex conjugate of their sum is given by the infimal convolution ψ * t (α) = min Since ψ Y i (s) = ψ t i X i (s) = ψ X i (t i s), by scaling property (3), one has Substituting α i = b i t i into (7) we obtain It means that Theorem 2.3 one can treat as the special case of a generalization of the infimal convolution to the situation of infinite numbers of summands.