Let $n$ and $k$ be positive integers. Let $X$ be the empirical mean of $n$ iid Rademacher random variables. Note that the distribution of $X$ is symmetric about 0, and also $|X| \le 1$ w.p 1. Let $X_1,X_2,\ldots,X_k$ be iid copies of $X$. Consider the random variable $Z_{n,k} := X_1X_2 \ldots X_k$.
Question.
- (1) If $k$ is fixed an $n \to \infty$, is there a large deviation principle for $Z_{n,k}$ with explicit rate function $I(x)$ ?
- (2) Same question when $k \asymp n \to \infty$.
- (3) Same question when $\log k \asymp n \to \infty$.
I came across this paper "Large deviation principle for random matrix products", but I find it difficult to read for a non-expert.
A related problem
If we instead define $Z_{n,k} = X^k$ with $k$ fixed and $n \to \infty$, then (apparently!) one has the large deviation inequality $\mathbb P(Z_{n,k} \ge a) \le 2e^{-n I(a^{1/k})}$, for all $a \in (0,1)$, with rate function $I(x) = \int_0^x \tanh^{-1}(t)\mathrm{d}t$, the ordinary / natural entropy. See the computation just before (2.5) of this manuscript http://www.numdam.org/article/AIHPB_2005__41_4_807_0.pdf
Update
One can obtain the following crude result. By standard Bernstein concentration, we have with $v=b=1/n$ it holds for any $\delta\in (0,1)$ that $$ P(|X_i| \ge \delta) \le 2e^{-\frac{\delta^2}{2(v+bt/3)}} = 2e^{-\frac{3 n\delta^2}{2(3+t)}} \le 2e^{-3n\delta^2/8}. $$
Thus, for any $a \in (0,1)$, one has $$ P(|Z_{n,k}| \ge a) \le \sum_{i=1}^k P(|X_i| \ge a^{1/k}) \le 2ke^{-3na^{2/k}/8}. $$
We deduce that if $n \to \infty$ such that $k \ge 3$ and $(\log k)/n \le c \lt 3/8$, then $$ \liminf -\frac{1}{n}\log P(|Z_{n,k}| \ge a) \ge \frac{3}{8}-c \gt 0, $$ for any $a \in (0,1)$.
