Question

Assume $0 < a_i \leq 1$ for $i = 1, 2 \ldots n$. I am interested in the random sum $X = \sum_i a_i X_i$ where $X_i$ are iid random Bernoulli variables with some mean $p \in (0, 0.5)$. I would like to know if one can relate $P(X \leq 1)$ to $P(X \leq \delta)$ for some $\delta < 1$. Specifically, what are the tightest bounds of the form

$$P(X \leq \delta) \geq f(\delta) P(X \leq 1)$$

for some function $f(\delta)$?

Answer 1

There is no such bound which depends only on $\delta$: if you take all $a_i=1$ and $p=1/2$, then for any $\delta<1$ the ratio between $\mathbb{P}(X\le \delta)$ and $\mathbb{P}(X\le 1)$ is $1/(n+1)$.