巴拿赫空间中的随机漫步及其函数

Bulletin of Mathematical Statistics Pub Date : 1978-03-01 DOI:10.5109/13125

M. Chang

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引用次数: 0

摘要

设an: n 1}是一个序列，其中E[X "]=0且Ell X.112<00的Banach空间值随机变量，设So=0, Sn= XiF X2+…+ Xn n 1。我们证明如果{Sn: n_。1}满足B中的LIL，则序列{77，，:n .1}满足C([0,1]， B)中的LIL，其中77n(t)=S[nt]+ (nt - [nt]) X[nt]-14, Ot51和C([°，1]，B)—={f:[0,1] ----。BI f是连续的。我们还利用这一结果给出了巴拿赫空间中布朗运动LIL的另一种证明。

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A RANDOM WALK AND ITS LIL IN A BANACH SPACE

Let an : n 1} be a sequence of i.i.d. Banach space valued random variables with E[X„]=0 and Ell X.112<00, and let So=0, Sn= XiF X2+ . . . + Xn, n 1. We prove that if {Sn : n_. 1} satisfies the LIL in B then the sequence {77,, : n .1} satisfies the LIL in C([0, 1], B), where 77n(t)=S[nt]+ (nt—[nt]) X[nt]-14, Ot51 and C([°, 1], B) --={ f : [0, 1] ----. BI f is continuous}. We also use this result to give an alternative to the proof of the LIL of Brownian motion in Banach spaces.

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Bulletin of Mathematical Statistics

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