# Monotonicity of $\mathbf{P} ( \bar{X}_N > 0 )$ in $N$

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.

The answer is, that $$p_N$$ is not necessarily increasing. Not that $$\mathbb{P}(\bar X_N > 0) = \mathbb{P}(X_1 + \ldots + X_N > 0)$$. Put $$\mathbb{P}(X=1) = 0.99$$ and $$\mathbb{P}(X=-98) = 0.01$$. Then $$\mathbb{E}X_1 = 0.01$$, but $$p_2 < p_1$$.
• Thank you, that makes sense. Do you get the sense that there would be any nontrivial assumptions one could make on $X$ which would cause it to be increasing? – πr8 Feb 12 at 17:10
• Only an idea: Assume that $X$ has symmetric distribution, i.e. $X \sim -X$ and $X_i \sim X + \mu$ with $\mu > 0$. – Dieter Kadelka Feb 12 at 17:16