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.
Publié le : 1980-06-14
Classification:
Mode,
median,
independent and identically distributed random variables,
limit theorem,
regularly varying tails,
60G50,
60F99
@article{1176994717,
author = {Hall, Peter},
title = {On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables},
journal = {Ann. Probab.},
volume = {8},
number = {6},
year = {1980},
pages = { 419-430},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994717}
}
Hall, Peter. On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables. Ann. Probab., Tome 8 (1980) no. 6, pp. 419-430. http://gdmltest.u-ga.fr/item/1176994717/