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Suppose that X n , n 1 and Y n , n 1 are both uniformly integrable sequences, and all the random
variables have finite second moments. (a) Show that if X n , Y n are independent then the
sequence Z n = X n Y n is uniformly integrable. (b) Will the statement remains true if we do not
assume independence? (Either prove the statement or provide a counterexample.)

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Suppose that Xn-n1 and Yn-n1 are both uniformly integrable sequences-.pdf

  • 1. Suppose that X n , n 1 and Y n , n 1 are both uniformly integrable sequences, and all the random variables have finite second moments. (a) Show that if X n , Y n are independent then the sequence Z n = X n Y n is uniformly integrable. (b) Will the statement remains true if we do not assume independence? (Either prove the statement or provide a counterexample.)