STAT 150: Stochastic Processes (Fall 2020)
Test #1
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1. Consider a Markov chain (Xn,n ≥ 0) on a finite state space S, where each
state i ∈ S is given a positive weight wi > 0. The Markov chain starts from a
uniformly random state X0 in S. At each subsequent step, the Markov chain
transitions according to the following two-step procedure. If Xn−1 = i, then
• first a state j 6= i is selected uniformly at random, and then,
• given this j, with probability w j/(wi+w j) the Markov chain transitions
to j (i.e., Xn = j), and with the remaining probability wi/(wi+w j) it
stays put (i.e., Xn = i).
Find the long run proportion of time spent in any given state i ∈ S. Fully
justify your answer.
2. Let α ∈ (0,1). Consider the Markov chain (Xn,n ≥ 0) on state space S =
{0,1,2, . . .} with transition probabilities
pi j =
{
(1−α)α j i= 0 and j ≥ 0
(1−α)α j−i+1 i≥ 1 and j ≥ i−1.
and all other pi j = 0. Using branching process theory, find the values of α for
which (Xn) is transient, null recurrent or positive recurrent. Fully explain your
answer. Hint: Relate the Markov chain (Xn) to a “slowed down” branching
process, where at any given step a single particle gives birth. Then use the
result of Problem 8(b) on Homework 2.
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