Approximation units and summation of independent random variables

AbstractLet {Xn} be a sequence of independent, identically distributed random variables. Assume that X1 has a density function with E(X1) = 0 and σn2(X1) < ∞. Call ƒn(x) the density of (1n) ∑i − 1n Xi. The Weak Law of Large Numbers gives ∝|x|>ϵƒn(x) dx → 0 for every ε > 0 whereas ∝ ƒn dx = 1. This tells us that ƒn ∗ g → g in the L1 metric whenever g ϵ L1(−∝, ∝). This can be readily seen in the case when g is continuous with compact support. The general case follows by a density argument as a consequence of Young's inequality. Throughout this paper we show that if in addition the characteristic function of X1 belongs to some class Lx, large α, then ƒn ∗ g converges a.e. to g. Similar results are discussed for the case when σ = ∞. It is shown that these results can be phrased in terms of a more general theorem concerning approximation units