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Invariance Theorems for First Passage Time Random Variables

Published online by Cambridge University Press:  20 November 2018

A. K. Basu
A. K. Basu*
Affiliation:
Laurentian University, Sudbury, Ontario
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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.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 2 , June 1972 , pp. 171 - 176
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Billingsley, P., Convergence of probability measures, Wiley, New York, 1968.Google Scholar
2. Heyde, C. C., Asymptotic renewal results for a natural generalisation of classical renewal theory, J. Roy. Statist. Soc. Series B, 29 (1967), 141-150.Google Scholar
3. Parthasarathy, K. R.,Probability measures on metric spaces, Academic Press, New York, 1967.Google Scholar