Let $\{a_i\}_{i=0}^{\infty}$ be a sequence of positive real numbers bounded above by some uniform constant. For each $k,n\geq 1$ let $$ A^{(k)}_n:=\prod_{i=k}^{n+k-1}a_i $$ and suppose that there exists $\lambda>0$ such that for some (and therefore for all) $k\geq 1$ we have \begin{equation} \liminf_{n\to\infty} A^{(k)}_n > \lambda>0. \end{equation} Then, for every $k\geq 1$ let $$ C^{(k)}:=\inf_{n\geq 1}\{A^{(k)}_n e^{-\lambda n}\}. $$ Notice that by the assumption we have that $C^{(k)}>0$.
Question 1: Is it true that $$\limsup_{k\to \infty} C^{(k)} > 0? $$
Question 2: does there exists $\epsilon>0$ such that $$ \liminf_{n\to\infty}\frac 1n \# \{1\leq k \leq n: C^{(k)}\geq \epsilon\}>0? $$
