Consider independent and identically distributed  random variables X₁, X₂, X₃, … where each X_i has a discrete uniform distribution on the integers 0, 1, 2, …, 9; that is, P(X_i = k) = 1/10 for k = 0, 1, 2, …, 9. Now form the sum

Intuitively, this is just the first n digits in the decimal expansion of a random number on the interval [0, 1]. Show that as n → ∞, Un converges in distribution to an rv U uniformly distributed on [0, 1], i.e. that P(U_n ≤ u) → P(U ≤ u), by showing that the moment generating function of U_n converges to the moment generating function of U.
[The argument for this appears on p. 52 of the article “A Few Counter Examples Useful in Teaching Central Limit Theorems,” The American Statistician, Feb. 2013.]

Central Limit Theorems

Transcribed Image Text:

n 1 Un Σ i=1 (10) = .1X + .01X2 + . . . = Χ +(.1)"Χn