By Peres Y.

Those notes checklist lectures I gave on the information division, collage 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 means 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 university in Jyvaskyla, August 1999.

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Then ψ is a bounded and continuous function on C[0, construction of {St }t≥0, we have Stn Stn max1≤k≤n Sk √ Eψ({ √ }0≤t≤1 ) = Eφ( max { √ }) = Eφ( ). 0≤t≤1 n n n Also, Eψ({B(t)}0≤t≤1) = Eφ( max B(t)). 0≤t≤1 Then, by Donsker’s Theorem, Eφ( max1≤k≤n Sk √ ) −→ Eφ( max B(t)). 3. 1) max{1 ≤ k ≤ n : Sk Sk−1 ≤ 0} =⇒ max{0 ≤ t ≤ 1|B(t) = 0} n The left hand side is the last time between 1 to n, scaled by n, that the random walk crosses 0. The right hand side is the last zero of Brownian motion in [0, 1].

Sm has a point of increase, and Nb = m). + m=n+1 Note that Nb ≤ n implies |B(b) − B(a)| ≤ (n + 1)h, so (n + 1)h P(Nb ≤ n) ≤ P(|B(b) − B(a)| ≤ (n + 1)h) = P |Z| ≤ √ , b−a where Z has a standard normal distribution. Since S0 , . 1 that for some constant C, we have ∞ P(S0 , . . , Sm has a point of increase, and Nb = m) m=n+1 ∞ ≤ P(Nb = m) m=n+1 C C ≤ . 1) can be made arbitrarily small by first taking n large and then picking h > 0 sufficiently small. 13. 3. For any random walk {Sj } on the line, P(S0 , .

Xd ) ∈ A. If A and B are increasing events, then it is easy to see by applying Harris’ Inequality to the indicator functions 1A and 1B that P(A ∩ B) ≥ P(A)P(B). 28 1. 3. Let X1 , . d. sample, where each Xi has distribution µ. Given any (x1 , . . , xn ) ∈ Rn , define the relabeling x(1) ≥ x(2) ≥ . . ≥ x(n) . Fix i and j, and define f (x1 , . . , xn) = x(i) and g(x1, . . , xn ) = x(j). Then f and g are measurable and nondecreasing in each component. Therefore, if X(i) and X(j) denote the ith and jth order statistics of X1 , .