Invariance Theorems for First Passage Time Random Variables
1972 ◽
Vol 15
(2)
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pp. 171-176
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Keyword(s):
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 thenconverges weakly to the Wiener measure W.Let τx(ω)=inf{k≥1|Sk>x}. In §2 we prove thatconverges 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.