MATH5835 – Assignment 1 Instruction: 1. This is individual assignment. The work should be your own work. 2. Total length: maximum of 3 pages (where one double-sided page counts as two pages). Note only the first 3 pages will be marked. 3. The assignment is due on Tuesday 10 March, 2020, at the beginning of the lecture. Late submission will result in a penalty. 1 [10 marks] Let X1, X2, … be i.i.d. r.v.’s with E(Xi) = µ and V ar(Xi) = σ 2 S2n = 1 n− 1 n∑ i=1 (Xi − X¯n)2. 1. Show that Sn P→ σ as n→∞. 2. Show that √ n(X¯n − µ) Sn D→ N(0, 1) as n→∞. 2 [10 marks] Consider a Markov chain with the transition probability matrix given by P = p0 1− p0 0 0 . . . p1 0 1− p1 0 . . . p2 0 0 1− p2 . . . … … … … … , where 0 < pk < 1, ∑∞ i=0 pi =∞. Which states are recurrent and which states are transient? 3 [10 marks] Let P be a transition matrix which is reversible with respect to the probability distribution pi on Ω. Show that the transition matrix P2 corresponding to two steps of the chain is also reversible with respect to pi. 1
