Invariance Theorems for First Passage Time Random Variables
- 1 June 1972
- journal article
- Published by Canadian Mathematical Society in Canadian Mathematical Bulletin
- Vol. 15 (2) , 171-176
- https://doi.org/10.4153/cmb-1972-031-9
Abstract
Let X1X2,… be i.i.d. r.v. with EX=μ>0, and E(X-μ)2 = σ2<∞.Let Sk=X1+…+Xk and vx=max{k:Sk≤x}, x≥0 and vx=0 if X1>x. Billingsley [1] proved if X1≥0 then converges weakly to the Wiener measure W.Let τx(ω)=inf{k≥1|Sk>x}. In §2 we prove that converges weakly to the Wiener measure when the X's may not necessarily be nonnegative. Also we indicate that this result can be extended to the nonidentical case.Keywords
This publication has 1 reference indexed in Scilit:
- PROBABILITY MEASURES IN A METRIC SPACEPublished by Elsevier ,1967