Suppose I have a sequence of i.i.d. random variables $X_1, X_2, \dots$ with positive mean $E[X_1] = \mu$. For $A>0$, is the function $$f(n) = \Pr( X_1 + X_2 + \dots + X_n \leq A) $$ monotonically non-increasing in $n$ (perhaps for $n$ large enough)?
Intuitively, I think this should be the case. By the law of large numbers, since $n\mu > A$ for $n$ large enough, $f(n) \to 0$ as $n \to \infty$, so there should be a monotonically decreasing subsequence of $f(n)$. But is the function actually monotonic for all $n$ (perhaps after some threshold)?