Let x x1 xn be a sample from p where 1 k with a fixed

Question

Let X = (X₁, ..., X_n) be a sample from P_θ, where θ ∈ {θ₁, ..., θ_k} with a fixed integer k. Let T_n(X) be an estimator of θ with range {θ₁, ..., θ_k}.

(a) Show that T_n(X) is consistent if and only if P_θ(T_n(X) = θ) → 1.

(b) Show that if T_n(X) is consistent, then it is an-consistent for any {a_n}.