Critical Behavior in Almost Sure Central Limit Theory

Let X ₁,X ₂,… be i.i.d. random variables with EX ₁=0, EX ₁²=1 and let S _k =X ₁+⋅⋅⋅+X _k . We study the a.s. convergence of the weighted averages where (d _k ) is a positive sequence with D _N =∑ _k=1 ^N d _k →∞. By the a.s. central limit theorem, the above averages converge a.s. to Φ(x) if d _k =1/k (logarithmic averages) but diverge if d _k =1 (ordinary averages). Under regularity conditions, we give a fairly complete solution of the problem for what sequences (d _k ) the weighted averages above converge, resp. the corresponding LIL and CLT hold. Our results show that logarithmic averaging, despite its prominent role in a.s. central limit theory, is far from optimal and considerably stronger results can be obtained using summation methods near ordinary (Cesàro) summation.