Let x(t) denote the number of
jumps occurring in the time interval [0,t) and vk(t) = P{x(t) = k}. The generating
function of vk(t) is given by
Lay off to the right of the origin successive intervals of length zj∕jα,j = 1,2,⋯ .
Explicitly the end points are
t1(z)
= 0
tj(z)
=∑k=1j−1zk∕kα,j = 2,3,⋯,α > 0,
and
t∞(z)
=∑k=1∞zk∕kα
Following Shepp and Lloyd Lr, the length of the r-th longest cycle and Sr, the length
of the r-th shortest cycle have been defined for our choice of x(t) and tj,j = 1,2,⋯ .
This paper obtains the asymptotics for the m-th moments of Lr and Sr suitably
normalized by a new technique of generating functions. It is further shown that
the results of Shepp and Lloyd are particular cases of these more general
results.
![∑∞ k ∑∞
exp{λt[ϕ(x) − 1]},ϕ(x) = pkx , pk = 1.
k=1 k=1](a030x.png)
