On Lerch's transcendent and the Gaussian random walk

Details On Lerch's transcendent and the Gaussian random walk On Lerch's transcendent and the Gaussian random walk

2007-03-30

arxiv.org/abs/math/0703908v1

Annals of Applied Probability 2007, Vol. 17, No. 2, 421-439

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051606000000781 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051606000000781

Let $X_1,X_2,...$ be independent variables, each having a normal distribution with negative mean $-\beta<0$ and variance 1. We consider the partial sums $S_n=X_1+...+X_n$, with $S_0=0$, and refer to the process $\{S_n:n\geq0\}$ as the Gaussian random walk. We present explicit expressions for the mean and variance of the maximum $M=\max\{S_n:n\geq0\}.$ These expressions are in terms of Taylor series about $\beta=0$ with coefficients that involve the Riemann zeta function. Our results extend Kingman's first-order approximation [Proc. Symp. on Congestion Theory (1965) 137--169] of the mean for $\beta\downarrow0$. We build upon the work of Chang and Peres [Ann. Probab. 25 (1997) 787--802], and use Bateman's formulas on Lerch's transcendent and Euler--Maclaurin summation as key ingredients.

Affiliated with

Also associated with

No associations

Rating

On Lerch's transcendent and the Gaussian random walk does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On Lerch's transcendent and the Gaussian random walk, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Lerch's transcendent and the Gaussian random walk will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-650989