Selected Topics in Probability and Stochastic Processes Steve Dunbar. Partial Converse of the Central Limit Theorem

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1 Steven R. Dunbar Department of Mathematics 203 Avery Hall University of Nebraska-Lincoln Lincoln, NE Voice: Fax: Selected Topics in Probability and Stochastic Processes Steve Dunbar Partial Converse of the Central Limit Theorem Rating Mathematically Mature: proofs. may contain mathematics beyond calculus with Question of the Day 1

2 Key Concepts Vocabulary Mathematical Ideas This section is adapted from: Larry Goldstein, A Probabilistic Proof of the Lindeberg-Feller Central Limit Theorem, American Mathematical Monthly, Volume 116, Number 1, January 2009, pp , particularly Section 4, page 57 This section is adapted from: Lemma 1 (Slutsky s Lemma) U n D U and V n P V implies U n +V n D U. Proof: Goldstein references T. Ferguson, A course in Large Sample Theory, Chapman and Hall, New York, 1996, but the proof appears to be fairly simple using the weak convergence equivalence to convergence in distribution, taking the set of test functions to say be the set of functions which are 2

3 continuously differentiable and have a bounded first derivative. Then the Mean-Value Theorem allows one to separate the functions and he boundedness of the derivative allows one to control the contribution made by the f (V n ) term by using the convergence in probability of the V n to the 0 random variable. Lemma 2 (Partial Converse to Slutsky s Lemma) Let U n and V n, n = 1, 2, 3,... be two sequences of random variables such that U n and V n are independent for each n. Then U n D U and U n + V n D U implies V n P V. Proof: Straightforward but detailed analysis and probability. Larry Goldstein, A Probabilistic Proof of the Lindeberg-Feller Central Limit Theorem, American Mathematical Monthly, Volume 116, Number 1, January 2009, pp Lemma 3 Let Y and Y n, n = 1, 2, 3,... be mean zero random variables with finite nonzero variances σ 2 = Var [Y ], and σn 2 = Var [Y n ] respectively. If Y D n Y and lim n σn 2 σ 2, then Yn Y Remark 1 This says that the zero bias transform has a continuity property under mild conditions. Proof: Theorem 4 (Partial Converse to CLT) Given a triangular array X j,n satisfying the standardization assumptions and also additionally that lim n m n = 0 where m n = max 1 j n σ 2 j,n. Then the small zero bias condition is necessary for W n D Z. Remark 2 Recall that the small zero bias condition is X I n,n p 0. Proof: Sketch: The continuity property of the zero bias transform implies that Wn D Z = Z. Since m n 0, a lemma from the proof of the CLT implies XI P n,n 0. Slutsky s Lemma now gives W n + X I n,n = W I n,n + X In,n D Z. Now W n is independent of I n and W n and therefore independent of X I n,n. Now use the partial converse to Slutsky s Theorem. 3

4 Problems to Work for Understanding 4. Reading Suggestion: Outside Readings and Links: 4

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