Consider the following autoregressive processes Wn 2Wn1 Xn

Consider the following autoregressive processes: Wn = 2Wn-1 + Xn w0 = 0 Zn = 3/4 Zn-1 + Xn Z0 = 0 Suppose that Xn is a Bernoulli process. What trends do the processes exhibit? Express Wn and Zn in terms of Xn, Xn-1,..., X1 and then find E[Wn] and E[Zn]. Do these results agree with the trends you expect? Do Wn or Zn have independent increments? stationary increments? Generate 100 outcomes of a Bernoulli process. Find the resulting realizations of Wn and Z. Is the sample mean meaningful for either of these processes? Repeat part d if Xn is the random step process.

Solution

a) Both the processes are exhibiting linear trend.

b)

X_n is a Bernoulli process.

So, E(X_i) = p

W₁ = 2W₀ + X₁ = X₁

W₂ = 2W₁ + X₂ = 2X₁ + X₂

W₃ = 2W₂ + X₃ = 2(2X₁ + X₂) + X₃ = 2²X₁ + 2X₂ + X₃

going in this process

W_n = 2^n-1X₁ + 2^n-2X₂ +...+ 2X_n-1 + X_n

So, E(W_n) = 2^n-1E(X₁) + 2^n-2E(X₂) +...+ 2E(X_n-1) + E(X_n) = p(2^n-1+2^n-2+...+2+1) = p(2ⁿ - 1)

_Z1 = 0.75Z₀ + X₁ = X₁

_Z2 = 0.75Z₁ + X₂ = 0.75X₁ + X₂

_Z3 = 2Z₂ + Z₃ = 0.75(0.75X₁ + X₂) + X₃ = 0.75²X₁ + 0.75X₂ + X₃

going in this process

Z_n = 0.75^n-1X₁ + 0.75^n-2X₂ +...+ 0.75X_n-1 + X_n

So, E(Z_n) = 0.75^n-1E(X₁) + 0.75^n-2E(X₂) +...+ 0.75E(X_n-1) + E(X_n) = p(0.75^n-1 + 0.75^n-2 + ... + 0.75 + 1)

= p(1 - 0.75ⁿ)/(1-0.75) = 4p(1 - 0.75ⁿ)

c) W_n or Z_n does not have independepent incerements but both have stationary incerement.