Convergence of Rice and Longuet-Higgins series for a Wong process

Let $\xi_t$ be a Wong process, i. e. a stationary Gaussian process with zero mean and the co-variance function
$$ p_t=\frac{3}{2}\exp\biggl(-\frac{|t|}{\sqrt 3}\biggr) \biggl[1-\frac{1}{3}\exp\biggl(-\frac{2}{\sqrt 3}|t|\biggr)\biggr]. $$
S. O. Rice and M. S. Longuet-Higgins used alternating series of factorial moments of the number of zeroes of $\xi_t$ for a representation of the distribution function $F_m(t)$ of the distance between the $i$ th and $(i+m+1)$th zeroes of $\xi_t$.
In the paper, the problem of convergence of these series is studied.