The Annals of Probability

Rounding of continuous random variables and oscillatory asymptotics

Svante Janson

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We study the characteristic function and moments of the integer-valued random variable ⌊X+α⌋, where X is a continuous random variables. The results can be regarded as exact versions of Sheppard’s correction. Rounded variables of this type often occur as subsequence limits of sequences of integer-valued random variables. This leads to oscillatory terms in asymptotics for these variables, something that has often been observed, for example in the analysis of several algorithms. We give some examples, including applications to tries, digital search trees and Patricia tries.

Article information

Ann. Probab., Volume 34, Number 5 (2006), 1807-1826.

First available in Project Euclid: 14 November 2006

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Zentralblatt MATH identifier

Primary: 60E05: Distributions: general theory 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability

Sheppard’s correction moments characteristic function Gumbel distribution random assignment digital search tree Patricia trie


Janson, Svante. Rounding of continuous random variables and oscillatory asymptotics. Ann. Probab. 34 (2006), no. 5, 1807--1826. doi:10.1214/009117906000000232.

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