Let $X_1, X_2, \cdots$ be independent and identically distributed random variables, and let $M_n$ and $m_n$ denote respectively the mode and median of $\Sigma^n_1X_i$. Assuming that $E(X^2_1) < \infty$ we obtain a number of limit theorems which describe the behaviour of $M_n$ and $m_n$ as $n \rightarrow \infty$. When $E|X_1|^3 < \infty$ our results specialize to those of Haldane (1942), but under considerably more general conditions.
"On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables." Ann. Probab. 8 (3) 419 - 430, June, 1980. https://doi.org/10.1214/aop/1176994717