Statistics Papers

Document Type

Journal Article

Date of this Version

1971

Publication Source

TheAnnals of Mathematical Statistics

Volume

42

Issue

3

Start Page

946

Last Page

951

DOI

10.1214/aoms/1177693323

Abstract

We find an explicit formula for the first passage probability, Qa(T|x) = Pr(S(t) < a, 0 ≦ t T | S(0) = x), for all T > 0, where S is the Gaussian process with mean zero and covariance ES(τ)S(t) = max (1-| t - τ|, 0). Previously, Qa(T | x) was known only for T ≦ 1.

In particular for T = n an integer and - ∞ < x < a < ∞,

Qa(T | x) = 1⁄φ(x) ∫D . . . ∫ det φ(yi - yj+1 + a) dy2 . . . dyn+1,

where the integral is a n-fold integral of y2, . . . , yn+1 over the region D given by

D = {a - x < y2 < y1 < . . . n+1}

and the determinant is of size (n + 1)x(n + 1), 0 < i, j n, with y0 ≡ 0, y1 ≡ a - x.

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Date Posted: 27 November 2017