# Question

The following is a sufficient condition for the central limit theorem: If the random variables X1, X2, . . . , Xn are independent and uniformly bounded (that is, there exists a positive constant k such that the probability is zero that any one of the random variables Xi will take on a value greater than k or less than – k), then if the variance of Yn = X1 + X2 + · · · + Xn becomes infinite when n. ∞, the distribution of the standardized mean of the Xi approaches the standard normal distribution. Show that this sufficient condition holds for a sequence of independent random variables Xi having the respective probability distributions

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