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 $n\to\infty$?
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, $\frac{Y_n}{\sigma\sqrt{n}} \Rightarrow N(0, 1)$. 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.
