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Let X = (X1, ..., Xn) be a sample from Pθ, where θ ∈ {θ1, ..., θk} with a fixed integer k. Let Tn(X) be an estimator of θ with range {θ1, ..., θk}.
(a) Show that Tn(X) is consistent if and only if Pθ(Tn(X) = θ) → 1.
(b) Show that if Tn(X) is consistent, then it is an-consistent for any {an}.
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