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# Entropy Estimates for the First Passage Time of a Random Walk to a Time Dependent Barrier

Anders Martin-Löf
Scandinavian Journal of Statistics
Vol. 13, No. 3 (1986), pp. 221-229
Stable URL: http://www.jstor.org/stable/4616028
Page Count: 9
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## Abstract

The first passage time of a random walk to a time dependent barrier is considered, $N(c)=\text{min}\{n,S_{n}>cb(n/c)\}$, where c is a large scale parameter. Large deviation estimates for the distribution of N(c) when C→∞ are derived saying that P(N(c) < ∞) ≈ $Ke^{-Rc}$, and that (N(c)-cT)/$\sqrt{c}$ is asymptotically gaussian given that N(c) < ∞. The parameters R and T are defined by a maximum entropy principle, which also provides the starting point of a heuristically natural proof of the limit theorem.

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