Sums of independent random variables-without moment conditions
Analogues of classical limit laws for sums of independent random variables (central limit theorem, laws of large numbers and law of the iterated logarithm) are discussed. We stress results which go through without moment or smoothness assumptions on the underlying distributions. These include (i) estimates for the spread of the distribution of Sn E2nXi in terms of concentration functions (Levy-Rogozin inequality), (ii) comparison of the distribution of Sn on different intervals (ratio limit theorems and Spitzer's theorem for the existence of the potential kernel for recurrent random walk), (iii) study of the set of accumulation points of Snlr(n) for suitable y(n) T oo. Only the following parallel to the law of the iterated logarithm is new: If X1, X2, * * * are independent random variables all with distribution F, Sn = jnX1, mn med (Sn), then there exists a sequence {y(n)} such that y(n) -0oo and -oo < lim inf (Sn mn)/T(n) < lim sup (Sn mn)/(n) < oo w.p. 1, if and only if F belongs to the domain of partial attraction of the normal law.