Answered: 6. Suppose that a random walk S₂ (in… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.8: Probability

Problem 32E

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6. Suppose that a random walk S, (in some graph/environment, such as Zd, but
not necessarily) is recurrent when started from position x. That is,
P(n ≥ 1, Sn=x|So = x) = 1.
Moreover, suppose that there is positive probability that the walk will eventu-
ally visit some other site y, when started from x. That is,
P(n ≥ 1, Sny|So = x) > 0.
Argue that, in fact, the random walk will visit y infinitely often, when started
from x.
Hint: Try a proof by contradiction, using the Law of Large Numbers.

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