The Annals of Probability

Random Stopping Preserves Regular Variation of Process Distributions

Priscilla Greenwood and Itrel Monroe

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Let $S_n$ be a stochastic process with either discrete or continuous time parameter and stationary independent increments. Let $N$ be a stopping time for the process such that $EN < \infty$. If the upper tail of the process distribution, $F$, is regularly varying, certain conditions on the lower tail of $F$ and on the tail of the distribution of $N$ imply that $\lim_{y\rightarrow\infty}P(S_N > y)/(1 - F(y)) = EN$. A similar asymptotic relation is obtained for $\sup_n S_{n \wedge N}$, if $n$ is discrete. These asymptotic results are related to the Wald moment identities and to moment inequalities of Burkholder. Applications are given for exit times at fixed and square-root boundaries.

Article information

Ann. Probab., Volume 5, Number 1 (1977), 42-51.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60J15 60J30

Stopping times random walk independent increments regular variation asymptotics boundary crossing maximum process


Greenwood, Priscilla; Monroe, Itrel. Random Stopping Preserves Regular Variation of Process Distributions. Ann. Probab. 5 (1977), no. 1, 42--51. doi:10.1214/aop/1176995889.

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