## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

Secondary: 60F99: None of the above, but in this section

**Keywords**

Mode median independent and identically distributed random variables limit theorem regularly varying tails

#### Citation

Hall, Peter. On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables. Ann. Probab. 8 (1980), no. 3, 419--430. doi:10.1214/aop/1176994717. https://projecteuclid.org/euclid.aop/1176994717