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In Bálint Tóth's paper, "No More Than Three Favourite Sites for Simple Random Walk", while proving one of the many technical lemmas in his theorem's proof, he makes the following claim: suppose for each $n\in\mathbb{N}$, $B_n$ is a binomially distributed random variable of $n$ trials and probability of success $p=\frac12$. That is, $P(B_n=\ell)={n\choose\ell}2^{-\ell}$ for any $\ell=0,1,\dots,n$. Then, for any $\gamma<\frac12$, $$\sup_n \operatorname{E}\left(\exp\{\gamma(2B_n-n)^2/n\}\right)<\infty.$$

I'm just not quite seeing why this has to be true, and Tóth never explains why it is. Maybe it's a well-known fact in probability theory, but I've never come across any instances of this precise result. Can it be proven with standard probability inequalities, or does it hinge on the fact that $B_n$ is binomial? Also, why exactly do we require $\gamma<\frac12$? Does there exist any $\gamma\geq\frac12$ such that this statement is still true?

I'll appreciate any help you have to offer.

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    $\begingroup$ The CLT shows that, when $n\to\infty$, $(2B_n-n)/\sqrt{n}$ converges in distribution to a standard normal random variable $Z$. Since the exponentia moment $E(e^{\gamma Z^2})$ is finite if and only if $\gamma<\frac12$, the result in Tóth's paper is not surprising and the supremum is indeed infinite when $\gamma\geqslant\frac12$. $\endgroup$
    – Did
    Nov 12, 2015 at 9:30

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