Let $X$ be a real-valued random variable satisfying
- $\mathsf E X = 0$ and
- $\mathsf P(X\le 0)\ge\frac12$.
In other words, $X$ is a (centralised) random variable whose median is below its mean. For instance, the distribution of wealth is (up to a constant) such a distribution: The median of income/net worth/etc. is always below the mean, since the mean is dragged up by a small number of very wealthy/high-income/etc. individuals.
Suppose that $X_1, X_2, \dots, X_n$ for some $n\in\mathbb N$ are all independent and identically distributed like $X$. Let $S_n\overset{\text{Def.}}=X_1+\dots+X_n$.
My reference request. Are there any known "nice" conditions on $X$ that are sufficient, so that
$$\mathsf P(S_n\le 0)\ge\frac12$$
holds? As was shown in this answer by hgmath, in general, the above inequality doesn't hold even if $n=2$.