随机游动局部时增量的若干结果

SOME RESULTS OF THE INCREMENT OF LOCAL TIME ON A RANDOM WALK

令N(t)表示一个周期性的随机游动局部时(到时间t为止达到点x的次数),我们有下列结果: (1) limsup N(T)/(T log log T)^(1/2)=2^(1/2)/σ a.s. (2) 如果 a_T/log T=∞则有σ(N(T)-N(T-t))/(f(log T/t+2log log t))^(1/2)=1 a.s.σ(N(s)-N(s-t))/(t(log T/t+2lon log t))^(1/2) a.s. σ(N(T)-N(T-S))/(t(log T/t+2 log log t))^(1/2)=1 a.s.

Let N(t) be the local time at zero (the number of returns to zero up to time t) of a reccurrent random walk. We obtain the following main theorem (1) (NT)/(T log log T)/^(1/2)=2^(1/2)/σ a. s. (2) a_T/log T=∞ then σ(N(T)-N(T-t))/(t(log T/t+2 log log t))^(1/2)=1 a. s. σ(N(s)-N(s-t))/(t(log T/t+2log log t))^(1/2)=1 a. s. σ(N(T)-N(T-s))/(t(t(log T/t+2 log log t))^(1/2)=1 a. s.