Question

4. Let Z1, Z2,... be a sequence of independent standard normal random variables. De- fine Xo 0 and n=0, 1 , 2, . . . . TL: n+1 , The stochastic process Xn,n 0, 1,2,3 is a Markov chain, but with a continuous state space. (a) Find EXn and Var(X). (b) Give probability distribution of Xn (c) Find limn oo P(X, > є) for any e> 0. (d) Simulate two realisations of the Markov process from n = 0 until n = 100. Provide your code and a picture.

Answer 1

Answer:

a) From the formula, we get n+ 1 Hence by recursion, TL i-0 as Xo-0. Now observe that Xn ! Zn V n E NU(0), as Xn-1-fCa , Z2, . . . , Ζ,1-1), where f is a linear function and Z's are independent of each other. Hence, TL .. By recursion we get, 72_ TL 2n+ 1 IL i-0

b) Since we have already seen that Xn is a linear combination of Z's hence X follows normal distribution. Now since we know it's mean and variance h N (Pmo;:) ie. Xn ~ N (0,1/(n + 1)) lence X n ~ 7L

Observe that Z-XnVn +1 ~ N (0, 1). Hence, c)

As per HomeworkLib policy,I have answered a,b,c parts