hw7 - Homework Set No. 7 Probability Theory (235A), Fall...

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Homework Set No. 7 – Probability Theory (235A), Fall 2009 Posted: 11/10/09 — Due: 11/17/09 1. (a) Read, in Durrett’s book (p. 63 in the 3rd edition) or on Wikipedia, the statement and proof of Kronecker’s lemma . (b) Deduce from this lemma, using results we learned in class, the following rate of con- vergence result for the Strong Law of Large Numbers in the case of a finite variance: If X 1 ,X 2 ,... is an i.i.d. sequence such that E X 1 = 0, V ( X 1 ) < , and S n = n k =1 X k , then for any ± > 0, S n n 1 / 2+ ± a.s. ---→ n →∞ 0 . Notes. When X 1 is a “random sign”, i.e., a random variable that takes the values - 1 , +1 with respective probabilities 1 / 2 , 1 / 2, the sequence of cumulative sums ( S n ) n =1 is often called a (symmetric) random walk on Z , since it represents the trajectory of a walker starting from 0 and taking a sequence of independent jumps in a random (positive or negative) direction. An interesting question concerns the rate at which the random walk
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hw7 - Homework Set No. 7 Probability Theory (235A), Fall...

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