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SECOND ORDER EXPANSIONS FOR THE MOMENTS OF

MINIMUM POINT OF AN UNBALANCED TWO-SIDED

NORMAL RANDOM WALK

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YANHONG WU

*Department of Mathematical Sciences, University of Alberta,*

Edmonton, AB, Canada T6G 2G1
(Received May 26, 1995; revised February 27, 1997)

**Abstract.**
In this paper, the second order
expansions for the first two moments of the minimum point of
an unbalanced two-sided normal random walk are obtained when
the drift parameters approach zero. The basic technique is
the uniform strong renewal theorem in the exponential family.
The comparison with numerical values shows that the
approximations are very accurate. It is shown, particularly,
that the first moment is significantly different from its
continuous Brownian motion analog while the second moments
are the same in the first order. The results can be used to
study properties of the maximum likelihood estimator for the
change point.

*Key words and phrases*:
Brownian motion, ladder
height and ladder epoch, strong renewal theorem.

**Source**
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