Invariance Theorems for First Passage Time Random Variables
Canadian mathematical bulletin, Tome 15 (1972) no. 2, pp. 171-176
Voir la notice de l'article provenant de la source Cambridge University Press
Let X 1X 2,... be i.i.d. r.v. with EX=μ>0, and E(X-μ)2 = σ2<∞.Let S k =X 1+...+X k and v x =max{k:S k ≤x}, x≥0 and v x =0 if X 1>x. Billingsley [1] proved if X1≥0 then converges weakly to the Wiener measure W.Let τx (ω)=inf{k≥1|S k >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.
Basu, A. K. Invariance Theorems for First Passage Time Random Variables. Canadian mathematical bulletin, Tome 15 (1972) no. 2, pp. 171-176. doi: 10.4153/CMB-1972-031-9
@article{10_4153_CMB_1972_031_9,
author = {Basu, A. K.},
title = {Invariance {Theorems} for {First} {Passage} {Time} {Random} {Variables}},
journal = {Canadian mathematical bulletin},
pages = {171--176},
year = {1972},
volume = {15},
number = {2},
doi = {10.4153/CMB-1972-031-9},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1972-031-9/}
}
[1] 1. Billingsley, P., Convergence of probability measures, Wiley, New York, 1968. Google Scholar
[2] 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] 3. Parthasarathy, K. R.,Probability measures on metric spaces, Academic Press, New York, 1967. Google Scholar
Cité par Sources :