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Invariance Theorems for First Passage Time Random Variables
Published online by Cambridge University Press: 20 November 2018
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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.
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- Copyright © Canadian Mathematical Society 1972
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