Does $\mathbb{P}(X_1 > a) \leq \delta$ imply $\mathbb{P}\left (\frac{1}{J} \sum_{i=1}^J X_i > a \right ) \leq \delta$?

Question

Let $X_1, X_2, ..., X_J$ be identically distributed random variables.

Edit: may assume independence as well as finite expectation of the $X_i$'s above.

Does $\mathbb{P}(X_1 > a) \leq \delta$ imply $\mathbb{P}\left (\frac{1}{J} \sum_{i=1}^J X_i > a \right ) \leq \delta$ ?

Answer 1

1: No, this isn't true. Let $Y$ be uniformly distributed on $\{1, 2, \dots, J\}$ and let $X_i = c \cdot 1_{\{Y = i\}}$ for some $c$. Then $P(X_1 > 0) = 1/J$ and $\frac{1}{J} \sum_{i=1}^J X_i = c/J$ a.s., so the claim is false if $0 < a < c/J$ and $1/J < \delta < 1$.