Mathematics · Statistical Inference

Let X1, X2, ..., Xn be a random sample from an N(θ, 1) distribution, where θ ∈ ℝ is unknown. Let δn be the Bayes estimator of θ under the squared error loss function L(θ, a) = (a - θ)2 and the prior distribution N(1, 2). If 1√n[(2n + 1)δn - 1 - 2nθ] converges in distribution to a random variable Z as n → ∞, then which of the following statements is true?