Stats 300 b : Theory of Statistics Winter 2018 Lecture 15 – February 27
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چکیده
Theorem 1. Let {Xn}n=1 ⊂ L∞(T ) be a sequence of stochastic processes on T . The followings are equivalent. (1) Xn converge in distribution to a tight stochastic process X ∈ L∞(T ); (2) both of the followings: (a) Finite Dimensional Convergence (FIDI): for every k ∈ N and t1, · · · , tk ∈ T , (Xn(t1), · · · , Xn(tk)) converge in distribution as n→∞; (b) the sequence {Xn} is asymptotically stochastically equicontinuous. Proof (1)⇒ (2) is trivial. Here we only prove (2)⇒ (1). Part I: Consider countable subsets of T . Let m ∈ N, and construct partitions Tm 1 , · · · , Tm km of T such that
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