*Suppose I iteratively add a given mean-preserving spread to a random variable. In the limit, will exactly half the mass be above $0$?*


Formally: Let $X$ be a random variable, and let $\varepsilon\_1,\varepsilon\_2,\ldots$ be i.i.d. random variables with strictly positive (but finite) variance and $E[\varepsilon\_i\mid X]=0$. Let $F\_n$ be the CDF of 
$$
X+\varepsilon\_1+\dots+\varepsilon\_n.
$$


Must it be the case that $F\_n(0)\to \frac{1}{2}$ as $n\to\infty$?



Limit of iterative addition of a mean-preserving spread
Note that $E[\varepsilon\_n] = E[E[\varepsilon\_n \mid X]] = 0$. Let $\sigma^2$ denote the variance of $\varepsilon\_n$.


Let $Y\_n = X + \varepsilon\_1 + \dots + \varepsilon\_n$. Note that $\frac{\varepsilon\_1 + \dots + \varepsilon\_n}{\sigma \sqrt{n}} \Rightarrow N(0,1)$ in distribution, by the central limit theorem, and $\frac{X}{\sigma\sqrt{n}} \to 0$ almost surely. So by [Slutsky's theorem](https://en.wikipedia.org/wiki/Slutsky%27s_theorem), $\frac{Y\_n}{\sigma\sqrt{n}} \Rightarrow N(0, 1)$. Then $P(Y\_n \le 0) = P(\frac{Y\_n}{\sigma \sqrt{n}} \le 0) \to \Phi(0) = \frac{1}{2}$ where $\Phi$ is the standard normal cdf.


