## The Annals of Probability

### On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables

Peter Hall

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 8, Number 3 (1980), 419-430.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994717

Digital Object Identifier
doi:10.1214/aop/1176994717

Mathematical Reviews number (MathSciNet)
MR573283

Zentralblatt MATH identifier
0442.60049

JSTOR