Let $X$ be a real-valued random variable with positive expectation (wlog, $\mathbf{E}[X] = 1$, say).

For $N \in \mathbf{N}$, let $X_1, \cdots, X_N$ be independent, identically-distributed copies of $X$, and let $\bar{X}_N = \frac{1}{N} \sum_{i =1}^N X_i$ be their sample mean. Now, consider the quantity

$$p_N \triangleq \mathbf{P} ( \bar{X}_N > 0 ) \in [0, 1].$$

**My question is:** Is it known whether $p_N$ is increasing with $N$?

Intuitively, it seems like it ought to be (edit, added after answer: I meant to say `*eventually*' here). If it can be proved with some moment assumption on $X$, I would also be happy with that, though it would be nice to do so without this assumption. A counter-example would also be interesting.