均值无限的随机游动上确界的尾渐近性和局部渐近性

Tail Asymptotics and Local Asymptotics for the Supremum of a Random Walk with an Infinite Mean

对于增量具有无限均值及长尾分布的随机游动,Denisov D.等给出了其上确界的尾渐近性的一个充分条件.本文将增量的长尾分布的范围扩大到一个更大的分布族,它真包含了长尾分布族和控制变化尾分布族等.同时,证明了上述的充分条件也是必要的.为此,研究了这个更大分布族的性质,给出了积分加权分布是长尾或次指数的一些充分条件.相应地,还得到了增量具有无限均值的随机游动上确界的局部渐近性的一个等价条件.

Denisov D. et al. delivered a sufficient condition on tail asymptotics of the supremum of a random walk with a common infinite mean and a long-tailed distribution of the summands. This paper cancels the restriction on the long-tail property of the distribution of the summands, thus enlarges the scope of distributions to a wider class, which includes properly long-tailed distribution class and dominatedly varying distribution class etc. Meanwhile, it is proved that the above sufficient condition is also necessary. In doing so, the authors investigate the properties of this wider class and provide some sufficient conditions on the integrated weighted distribution of the summands to be long-tailed or subexponential. Correspondingly, the authors also give an equivalent condition on local asymptotics of the supremum of a random walk with an infinite mean of the summands.