Topics

- Sections: 3.9, 3.10, 5.3, 5.4, 6.7
- Definitions:
- Simple random walk and notation
- Asymptotics of Pr[S_n = k]
- Galton Watson branching process
- Extinction / Extinction Probability
- Generating function of Z_n

- Theorems: 5.4.1, 5.4.2, 5.4.5, 6.7.1, 6.7.7, 6.7.8
- Sample Questions:
- What is the expectation of S_1 given S_n = k?
- What is the variance of S_1 given S_n = k?
- 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))
- Prove that if the mean offspring size is < 1, then a branching process goes extinct almost surely.
- Say the offspring distribution is Poisson with mean 1. What is the variance of Z_n?
- Say two offspring distributions both have mean 2, but the first has variance 1 and the second variance 2. Does one or the other have a hihger likelihood of eventually going extinct? Or is there not enough information?
- Say the offspring distribution has mean mu and variance 1. What is the covariance of Z_n and Z_m?
- Prove that if mu >1 there is a positive probability of never going extinct.
- For a branching process with poisson(lambda) offspring distribution find the best asymptotics you can for the probability that the process goes extinct at step n
- Poisson branching process with mean lambda <1 (and for lambda =1), find the asymptotics for the process surviving until at least step n
- Derive the equation for the survival probability of a Poisson branching process with mean 2
- Let U be uniform [0,1], and then let Z_n be a branching process with offspring distribution Bin(2, U). What is E(Z_n)?