How small are the increments of partial sums of non-i.i.d. random variables (Q1201631)

Let \(X_ 1,X_ 2,\dots\) be a sequence of independent r.v.'s with \(\mathbb{E} X_ i=0\), satisfying some weak moment conditions. Further let \(X_ n+X_{n+1}+\dots+X_{n+k}=S(n,k)\). The author is interested in the limit behaviour of \(T_ N=\min_{0\leq n\leq b_ N} \max_{0\leq k\leq a_ N} S(n,k)\) where \(\{a_ N\}\) and \(\{b_ N\}\) are sequences of positive integers satisfying some regularity conditions. One of his main results tells us that \(\liminf_{N\to\infty} \gamma_ N T_ N=1\) a.s. where the normalizing factor \(\gamma_ N\) depends on the sequences \(\{a_ N\}\), \(\{b_ N\}\), \(\{\mathbb{E} X_ N^ 2\}\).