Convergence of the Longuet-Higgins series for Gaussian stationary Markov process of the first order

Let $\biggl(\xi_t,\frac{d\xi_t}{dt}\biggr)$ be a Gaussian stationary Markov process. M. S. Longuet-Higgins used alternating series (coefficients of which are expressed in terms of factorial moments of the number of zeroes of $\xi_t$) for a representation of the distribution function of the distance between the $i^{th}$ and the $(i+m+1)^{th}$ zeroes of $\xi_t$. In this paper the problem of convergence of these series is studied.