Local limit theorems for the density of the maximum of sums of independent random variables (Q1109407)

Let \(\xi\), \(\xi_ 1\), \(\xi_ 2\), \(\cdot \cdot \cdot\) be a sequence of random variables with common density function p(x). Suppose \(a=E \xi >0\), \(\sigma\) \(2=E(\xi -a)\) \(2=1\). Let \(S_ n=\xi_ 1+\cdot \cdot \cdot +\xi_ n\), \[ \bar S_ n=\max_{1\leq k\leq n}S_ k,\quad Z_ n=(S_ n-an)/\sigma \sqrt{n},\quad \bar Z_ n=(\bar S_ n-an)/\sigma \sqrt{n}, \] and let \(\Delta_ n(x)=| p_ n(x)-\Phi (x)|\), \({\bar \Delta}{}_ n(x)=| q_ n(x)-\Phi (x)|\), where \(p_ n(x)\) and \(q_ n(x)\) are density functions of \(S_ n\) and \(\bar S_ n\), respectively, and \(\Phi (x)=(2\pi)^{-1/2}\exp (-x\) 2/2). The author proves that if p(x)\(\leq C\), then \[ {\bar \Delta}_ n(x)\leq \Delta_ n(x)+LC\quad 2\ln n/(a\sqrt{n})+L(1+C\quad 4)/(a\quad 2\sqrt{n}), \] where L is an absolute constant. This estimate is improved when \(E| \xi |\) \(3<\infty\).