The Annals of Probability

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

Peter Hall

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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


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