What is the probability that two independent random walks both starting from 0 are equal at step k? Find the asymptotics of this probability
Find the asymptotics of the probability that S_n = 0 (for even n)
What is the expectation of S_1 given that S_100 = 0 and S_200 = 50?
Let S_n = X_1 + ... + X_n be a simple symmetric random walk, and let Y_i = X_i with probability p,and Y_i= - X_i with probability 1-p, and T_n = Y_1 + ... + Y_n. What is the distribution of T_n? What is the covariance of S_n and T_n?
Show that with high probability, the random walk does not hit zero anytime between step (n- n^(1/3)) and step (n + n^(1/3))