Sufficient conditions for monotonicity with probability distributions

Question

Let $X_i$ be a continuous non-negative real-valued random variable and $i=1,...,n$. $X_i$ are not necessarily independent over $i$. Let $b>0$, $\delta>0$. Consider $$ A_n:=\operatorname{Pr}\left(\delta\sum_{i=1}^{n}X_i>b\right) $$ Do you know some sufficient conditions on the distribution of each $X_i$ such that $A_n$ is monotone increasing (decreasing) in $n$ when $\delta>0$ ($\delta<0$)?

Answer 1

$A_n$ is increasing in $n$ because the event $$ \bigg\{ \delta \sum_{i=1}^{n-1} X_i > b \bigg\} $$ is a subset of the event $$ \bigg\{ \delta \sum_{i=1}^n X_i > b \bigg\}, $$ simply because $X_n$ is nonnegative.