Some Limit Theorems in Log Density

Motivated by recent results on pathwise central limit theorems, we study in a systematic way log-average versions of classical limit theorems. For partial sums S(k) of independent r.v.'s we prove under mild technical conditions that (1/log N)SIGMA(k less-than-or-equal-to N)(1/k)I{S(k)/a(k) is-an-element-of .} --> G(.) (a.s.) if and only if (1/log N)SIGMA(k less-than-or-equal-to N)(1/k)P(S(k)/a(k) is-an-element-of .) --> G(.). A functional version of this result also holds. For partial sums of i.i.d. r.v.'s attracted to a stable law, we obtain a pathwise version of the stable limit theorem as well as a strong approximation by a stable process on log dense sets of integers. We also give necessary and sufficient conditions for the law of large numbers in log density.