I am trying to prove an inequality that seems to be intuitively true, however I cannot arrive at a rigorous argument.
Consider a sequence of i.i.d random variables $X_1,X_2,....$, that take values in $[0,\infty)$ such that $\mathbb{E}[X_i] = \mu$. Let $\beta_1 > \beta_2 \geq 0$. Suppose, \begin{align*} \mathbb{E}[X_i\mathbb{1}\{X_i < \beta_1\}] &= \mu_1 \\ \mathbb{E}[X_i\mathbb{1}\{X_i < \beta_2\}] &= \mu_2 \end{align*}
Further consider the sums, \begin{align*} S_n &= \sum_{i=1}^{n} X_i\mathbb{1}\{X_i < \beta_1\} \\ S_l &= \sum_{i=1}^{l} X_i\mathbb{1}\{X_i < \beta_2\} \end{align*} such that $n > l$.
Is the following inequality true:
\begin{align*} \mathbb{P}(S_n > n (\mu_1 + \gamma_1) \vert S_l \leq l (\mu_2 + \gamma_2)) \leq \mathbb{P}(S_n > n (\mu_1 + \gamma_1)) \end{align*} where $\gamma_1 < \gamma_2$.
The objective is to use i.i.d Chernoff bounds to bound the term of the l.h.s even though dependence is introduced by the conditioning.
