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Miklós Schweitzer 2007 Problem 10

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March 3, 2017
Miklos Schweitzerprobability

Problem Statement

Let ζ1,ζ2,…\zeta_1, \zeta_2,\ldots be identically distributed, independent real-valued random variables with expected value 00. Suppose that the Λ(λ):=log⁡Eexp⁡(λζi)\Lambda (\lambda) := \log \mathbb E \exp (\lambda \zeta_i) logarithmic moment-generating function always exists for λ∈R\lambda\in\mathbb R (E\mathbb E is the expected value). Furthermore, let G ⁣:R→RG\colon\mathbb R \rightarrow \mathbb R be a function such that G(x)≤min⁡(∣x∣,x2)G(x)\leq \min (|x|, x^2). Prove that for small γ>0\gamma >0 the following sequence is bounded: {Eexp⁡(γlG(1l(ζ1+…+ζl)))}l=1∞\left\{ \mathbb E \exp \left( \gamma l G \left( \frac 1l (\zeta_1+\ldots + \zeta_l)\right)\right)\right\}^{\infty}_{l=1}
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