Strong laws for small increments of renewal processes (Q1180522)

Let \((N(t), t\geq 0)\) be a renewal process generated by a sequence of independent identically distributed random variables \(X_ 1,X_ 2,\ldots\) with \(0<E X_ 1<\infty\). Strong limit theorems are obtained for the increments \(\sup_{0\leq t\leq T-K}(N(t+K)-N(t))\), where \(K=K_ T\) is a function of \(T\) such that \(K_ T\uparrow\infty\), but \(K_ T/\log T\downarrow 0\) as \(T\to\infty\). The cases of \(P(X_ 1=0)>0\) and \(P(X_ 1=0)=0\) are considered separately.