Let (X_n)^\infty_{n=1} be a sequence of A's and B's formed as follows. The first two letters, X_1 and X_2, are chosen independently and at random, with Pr(A) = Pr(B) = 1/2. For n \geq 2 (i.e. n>=2), letter X_{n+1} is selected in a way that depends on previously selected letters.

(1). Suppose X_{n+1}, for n \geq 2 (i.e. n>=2) , is selected conditionally on X_n with

Pr(X_{n+1} =A|X_n =A)=1/2

Pr(X_{n+1} =A|X_n =B)=1/4

Find the proportion of A's and B's in an infinitely long sequence (X_n) using ergodic theorem with proper stationary distribution.

(2). Suppose we instead select X_{n+1}, for n \geq 2, conditionally on the preceding two letters with

Pr(X_{n+1} = A|X_n = A,X_{n-1} = A) = Pr(X_{n+1} = A|X_n = B,X_{n-1} = A) = 1/2

Pr(X_{n+1} = A|X_n = A,X_{n-1} = B) = Pr(X_{n+1} = A|X_n = B,X_{n-1} = B) = 1/4

(X_{2n-1})^\infty _{n=1} is a Markov chain if we denote odds and even numbered terms separately into two Markov Chains.

In the setting from (2), find the proportion of A's and B's in an infinitely long sequence using ergodic theorem with proper stationary distribution.

(See more details of the equations and sequences in the snapshot. Same content)

Let (Xn). n=1 be a sequence of A's and B's formed as follows. The first two letters, X1 and X2, are chosen independently and at random, with Pr(A) = Pr(B) = 1/2. For n 2 2, letter Xn+1 is selected in a way that depends on previously selected letters. (1). Suppose Xn+1, for n 2 2, is selected conditionally on Xn with Pr(Xn+1 = AXn = A) = 1/2 Pr(Xn+1 = AXn = B) = 1/4 Find the proportion of A's and B's in an infinitely long sequence (Xn) using ergodic theorem with proper stationary distribution. (2). Suppose we instead select Xn+1, for n 2 2, conditionally on the preceding two letters with Pr(Xn+1 = AXn = A, Xn-1 = A) = Pr(Xn+1 = A[Xn = B, Xn-1 = A) = 1/2 Pr(Xn+1 = AXn = A, Xn-1 = B) = Pr(Xn+1 = AXn = B, Xn-1 = B) = 1/4 (Xzn-1)=1 a Markov chain, if we denote odds and even numbered terms separately into two Markov Chains. In the setting from (2), find the proportion of A's and B's in an infinitely long sequence using ergodic theorem with proper stationary distribution