University of Washington Math 523A
Final exam
Due by: June 9
Answer exactly 4 of the following questions:
1. Let Stbe a simple random walk on Zand set τ= min{t:|St| ≥ a}for some integer a > 0.
Calculate Eτ2.
Hint : Consider a martingale involving the fourth power of St.
2. Consider a biased random walk on the integers: for some 0 <p<1,
St+1 =St+ 1 p
St−1 1 −p.
Let τa= min{t:St=a}, and let a, b be two positive integers.
(a) Calculate P(τ−a< τb).
(b) Compute E[min{τ−a, τb}].
3. In a sequence of fair coin tosses, let τ011 and τ000 denote the number of tosses until witnessing
the corresponding pattern.
(a) Find the value of Emax{τ011, τ000}.
(b) Find Eτ2
01.
4. A gambler plays the following game. In each round, he can pay any 0 < p < 1 dollars, and
win $1 with probability p(independently). Show that the probability that the gambler’s net
gain exceeds hat any of the first nrounds is at most exp(−h2/2n).
5. Consider percolation on a binary tree with parameter p < 1
2. Let Cdenote the cluster of the
root. Show that
P(|C| > k)≤exp(−kα(p))
for some α(p)>0 (find α(p) explicitly).
6. Let Stbe a simple random walk on Z, and for c, α > 0 define
τc,α = min{t:|St| ≥ ctα+ 1}.
(a) For which c, α is τc,α <∞almost surely?
(b) For which c, α is Eτc,α <∞almost surely?