Cramér transform of Rademacher series

A variational formula for the Cramér transform of series of weighted, independent symmetric Bernoulli random variables (Rademacher series) is given.

is called the Legendre-Fenchel transform (convex conjugate) of f and a function f * * : X → R ∪ {∞} defined by is called the convex biconjugate of f . The functions f * and f * * are convex and lower semicontinuous in the weak* and weak topology on X * and X , respectively. Moreover, the biconjugate theorem states that the function f : X → R ∪ {∞} not identically equal to +∞ is convex and lower semicontinuous if and only if f = f * * .
Let I be a countable set and ( i ) i∈I be a Bernoulli sequence, i.e. a sequence of i.i.d. symmetric r.v's taking values ±1. For t = (t i ) i∈I ∈ 2 (I ) ≡ 2 the series X t := i∈I t i i converges a.s.. Notice that for t ∈ 1 i.e. X t is a bounded r.v. and we can define its cumulant generating function on whole R that is Observe that We can not derive an evident form of ψ * t by using the classical Legendre transform because we can not solve (inverse the derivative ψ t ) the equation and find where s α is a solution of the Eq. (1).
The following theorem shows some variational expression on ψ * t .
Theorem 1.1 Let ( i ) i∈I be a Bernoulli sequence and t = (t i ) i∈I ∈ 1 (I ). The Cramér transform of a variable X t = i∈I t i i is given by the following variational formula is the convex conjugate of a functional ψ 1 : 1 → R of the form ψ 1 (t) = ln Ee X t and D(ψ * 1 ) ⊂ ∞ (I ) denotes its effective domain.
Remark 1.1 Presented in the next section proof techniques are similar, but not the same, to methods used by Ostaszewska and Zajkowski in [6,7].

Proof of Theorem 1.1
We begin with an observation on the absolute value of the cumulant generating function: |ψ t (s)| ≤ |s| t 1 . A parameter t may be an arbitrary element of 1 . Formally we can define a function ψ of two variables: Fixing t or s we write ψ(s, t) = ψ t (s) or ψ(s, t) = ψ s (t), respectively. First we derive ψ * s and next we show how ψ * t is expressed by ψ * s . In a standard way one can check the convexity of ψ s for every s ∈ R. Let t, u ∈ 1 and λ ∈ (0, 1) then Using the Hölder inequality for exponents 1/λ and 1/(1 − λ) we get E e s i∈I t i i λ e s i∈I u i i 1−λ ≤ Ee s i∈I t i i λ Ee s i∈I u i i 1−λ and, in consequence, Let a = (a i ) i∈I ∈ ∞ . By the definition of the convex conjugate we have where t, a = i∈I t i a i . Note that for s = 0 we have Assume now that s = 0. An expression in the curly bracket of (2), denote it by w, is concave and its partial derivatives along vector of basis e i = (δ i j ) j∈I in 1 (δ i j is the Kronecker delta) equal The expression w is a sum of functions with separated variables (t i ) i∈I . Concavity of each of these functions implies that the gradient ∇w(t) = (a i − s tanh(st i )) i∈I belongs to the subgradient ∂w(t) since The concave function w attained its maximum (global) at the point t if and only if 0 ∈ ∂w(t). It suffices that Because arc tanh(x) = 1 2 ln 1+x 1−x for |x| < 1 then the partial derivatives equal zero when Substituting the above values of t i 's into (2) we get Look a bit closely at the effective domain of ψ * s that is at the set is even and f (0) = 0. Since lim |x|→1 − = 2 ln 2 we can extend its domain to the interval [−1, 1]. One can check that and |a i | ≤ |s|. Let B ∞ (0; r ) denote of the closed ball at the center 0 and radius r in the space ∞ . The properties of f gives that Let us note that D(ψ * s ) is a symmetric set that is a ∈ D(ψ * s ) if and only if −a ∈ D(ψ * s ). Moreover it is symmetric with respect to each coordinates a i of a.
Return to the function ψ t . Let us observe that . Because ψ t is convex and continuous on R then, by the biconjugate theorem, we get On the other hand If we take a = sb then ψ * s (sb) = ψ * 1 (b) with b ∈ D(ψ * 1 ). It means that we can rewrite the above variational principle as follows We show that every number in (− t 1 , t 1 ) is taken by the inner product t, b over the set D(ψ * 1 ). Observe that a vector b = i∈J r (sgn t i )e i , where J is some finite subset of I and r ∈ [−1, 1], belongs to D(ψ * 1 ) (only finite number of nonzero terms). For this vector we have It follows that the inner product t, b attains over the set D(ψ * 1 ) any number belonging to the interval (− t 1 , t 1 ).
. Now we can divide the supremum of (3) into two parts and get Define a function We prove that in the above definition of function ϕ t an infimum over the set D(ψ * 1 )∩ {b ∈ ∞ : t, b = α} is attained and we can replace it by a minimum over this set that is we prove for α ∈ (− t 1 , t 1 ) and +∞ otherwise.
By Banach-Alaoglu theorem the closed (unit) ball B ∞ (0; 1) ⊂ ∞ ( 1 ) * is weak* compact and for each t and α ∈ (− t 1 , t 1 ) the hyperplain H t,α = {b ∈ ∞ : t, b = α} is closed in this topology. We have that an intersection B ∞ (0; 1) ∩ H t,α is weak* compact. Let 0 be the space of sequences with finite support. Obviously 0 ∩ B ∞ (0; 1) ⊂ D(ψ * 1 ) and H t,α ∩ 0 = ∅. We have Recall that the function ψ * 1 is nonegative and lower semicontinuous in the weak* topology. By Weierstrass Theorem ψ * 1 attains its minimum in the compact set B ∞ (0; 1) ∩ H t,α . Because an intersection of this set with the effective domain of ψ * 1 is nonempty then it means that a nonegative infimum is attained at some element in D(ψ * 1 ). It follows that in the definition of ϕ t we can replace the infimum by minimum and the formula (5) holds.
The formula (4) means that ψ t is the convex conjugate of ϕ t . To prove an equality ϕ t = ψ * t we should show that ϕ t is convex and lower semicontinuous. First we check the convexity of ϕ t . Take α 1 , α 2 ∈ (− t 1 , t 1 ). If α 1 or α 2 do not belong to the interval (− t 1 , t 1 ) then the value of ϕ t at such α k equals ∞ and the condition of convexity is trivially satisfied. Let b k (k = 1, 2) be vectors in Observe that for λ ∈ (0, 1) The above and convexity of ψ * 1 gives Now we prove the lower semicontinuity of ϕ t . Recall that ψ * 1 is convex and lower semicontinuous in the weak* topology on ∞ . It means that for any c ∈ R the set is weak* closed. Since ψ * 1 ≥ 0 we can assume that c ≥ 0. Because the above set is contained in weak* compact unit ball B ∞ (0; 1) ⊃ D(ψ * 1 ) then it is also compact in this topology. Consider a range of the set (6) by the functional l t := t, · , i.e.
Since for each t ∈ 1 the linear functional l t is continuous on ∞ (also in the weak* topology), by the intermediate and extreme value theorems we get that the set (7) is a closed interval. By symmetry of the set (6) and linearity of the functional l t we get the existence of a real number α such that We show that Let β ∈ ϕ −1 t ((−∞, c]). Since ψ * 1 is lower semicontinuous, there exists b β such that .

Remark 2.1
The result of Theorem 1.1 is similar to those obtained by the contraction principle (see for instance [3]) but let us emphasize that we used the space of parameters 1 to generate the convex conjugate of the investigated function and we did not consider any probability distribution on it.
Remark 2.2 Let us stress that the proof of Theorem 1.1 contains some scheme which allow us to generate, under some assumptions of course, variational formulas on the Cramér transform for another series of random variables.
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