An exact asymptotic for the square variation of partial sum processes

We establish an exact asymptotic formula for the square variation of certain partial sum processes. Let {X-i} be a sequence of independent, identically distributed mean zero random variables with finite variance sigma(2) and satisfying a moment condition E [| X-i vertical bar(2+delta)] < infinity for some delta > 0. If we let P-N denote the set of all possible partitions of the interval [N] into subintervals, then we have that max(pi is an element of PN) Sigma(I is an element of pi) Sigma(I is an element of pi) vertical bar Sigma i is an element of I X-i vertical bar(2) similar to 2 sigma N-2 ln ln(N) holds almost surely. This can be viewed as a variational strengthening of the law of the iterated logarithm and refines results of J. Qian on partial sum and empirical processes. When delta = 0, we obtain a