By Peres Y.

Those notes checklist lectures I gave on the information division, college of California, Berkeley in Spring 1998. i'm thankful to the scholars who attended the direction and wrote the 1st draft of the notes: Diego Garcia, Yoram Gat, Diogo A. Gomes, Charles Holton, Frederic Latremoliere, Wei Li, Ben Morris, Jason Schweinsberg, Balint Virag, Ye Xia and Xiaowen Zhou. The draft was once edited by way of Balint Virag, Elchanan Mossel, Serban Nacu and Yimin Xiao. I thank Pertti Mattila for the invitation to lecture in this fabric on the joint summer time institution in Jyvaskyla, August 1999.

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Xn, then it follows from Harris’ Inequality that E[X(i)X(j)] ≥ E[X(i)]E[X(j)], provided these expectations are well-defined. See Lehmann (1966) and Bickel (1967) for further discussion. For the rest of this section, let X1 , X2 , . . d. random variables, and let Sk = Xi be their partial sums. Denote k i=1 pn = P(Si ≥ 0 for all 1 ≤ i ≤ n) . 3) Observe that the event that {Sn is the largest among S0 , S1, . . Sn } is precisely the event that the reversed random walk Xn + . . + Xn−k+1 is nonnegative for all k = 1, .

However, for d = 1, suppose A = ZB = {t : B1 (t) = 0}. We have shown that dimH (ZB ) = 1/2 almost surely, but dimH (B1 (ZB )) = dimH ({0}) = 0. 3. Let α < dimH (A). 2, there exists a Borel probability measure µ on A such that Eα (µ) < ∞. Denote by µB the random measure on Rd defined by µB (D) = µ(Bd−1 (D)) = µ({t : Bd (t) ∈ D}) for all Borel sets D. Then E[E2α(µB )] = E Rd Rd dµB (x)dµB (y) =E |x − y|2α R R dµ(t)dµ(s) , |Bd (t) − Bd (s)|2α where the second equality can be verified by a change of variables.

Sn has a point of increase) ≤ 2 n k=0 pk pn−k n/2 2 k=0 pk . 2) Proof. The idea is simple. 3), and given that there is at least one such point, the expected number is bounded below by the denominator; the ratio of these expectations bounds the required probability. To carry this out, denote by In (k) the event that k is a point of increase for S0 , S1 , . , Sn and by Fn (k) = In (k) \ ∪k−1 i=0 In (i) the event that k is the first such point. The events that {Sk is largest among S0 , S1 , .