UDC 519.214.3

MATHEMATICS

Yu. K. BELYAEV

A LIMIT THEOREM FOR THE NUMBER OF CROSSINGS OF A HIGH LEVEL BY A STATIONARY GAUSSIAN PROCESS

(Presented by Academician A. N. Kolmogorov on 28 V 1966)

Let \(\xi_t\) be a stationary Gaussian process, \(\mathbf M \xi_t = 0\), \(\mathbf M \xi_s \xi_{s+t} = \rho(t)\), \(-\infty < t < +\infty\). It is said that the process \(\xi_t\) crosses the level \(u\) from below upward at the time \(\tau\) if \(\xi_\tau = u\), and in a sufficiently small neighborhood \((\tau-\varepsilon,\tau+\varepsilon)\) of the point \(\tau\), \(\varepsilon=\varepsilon(\omega)\) being a random variable, \(\xi_s<u\), \(\tau-\varepsilon<s<\tau\), \(\xi_s>u\), \(\tau<s<\tau+\varepsilon\). Denote by \(\eta_u(\Delta)\) the number of crossings from below upward of the level \(u\) on the interval \(\Delta\). Denote by \(\widetilde\eta_u(\Delta)\) the accompanying random variable, taking integer values \(\widetilde\eta_u(\Delta)=\eta_u(\Delta)\) if in the interval \(\Delta\) there are no crossings following one another after a time less than \(\tau\); otherwise we put \(\widetilde\eta_u(\Delta)=0\).

Everywhere below it is assumed that the correlation function \(\rho(t)\) is twice differentiable and satisfies the conditions \(\rho(0)=1\),

\[ |\rho''(t)-\rho''(0)| \leq c/|\ln |t||^{1+\varepsilon},\qquad c,\varepsilon>0,\qquad t\to 0; \tag{1} \]

\[ \rho(t)=o(1/\ln t),\qquad \rho'(t)=o(1/\sqrt{\ln t}),\qquad t\to\infty. \tag{2} \]

As the level \(u\) is increased \((u\to\infty)\), we shall increase \(|\Delta|\)—the length of the interval \(\Delta\)—in such a way that \(e^{-u^2/2}|\Delta|=\mathrm{const}\). For brevity we denote such a coordinated increase of \(u\) and \(|\Delta|\) by \((u,|\Delta|)\to\infty\). Denote by \(A_u(|\Delta|,\tau)\) the event consisting in the fact that on the interval \(\Delta\) there are crossings of the level \(u\) from below upward which follow one another after a time less than \(\tau\).

Theorem 1. If the correlation function of the Gaussian process \(\xi_t\) satisfies (1), then for \(\tau\leq v_u=\exp\{o(u^2)\}\), \(u\to\infty\),

\[ \lim_{(u,|\Delta|)\to\infty}\mathbf P\{A_u(|\Delta|,\tau)\}=0. \tag{3} \]

In the proof of the theorem the limiting relation

\[ \lim_{u\to\infty}\frac1\tau e^{u^2/2} \int_0^{2\tau}\int_0^{2\tau} \mathbf M\bigl(\dot\xi_{t_1}^{+}\dot\xi_{t_2}^{+}\mid \xi_{t_j}=u,\ j=1,2\bigr) p_{t_1,t_2}(u,u)\,dt_1\,dt_2=0, \]

is used, where \(p_{t_1t_2}(x_1,x_2)\) is the joint probability density of the values \(\xi_{t_1},\xi_{t_2}\), and \(\dot\xi_t^{+}=\dot\xi_t\) if \(\dot\xi_t>0\); \(\dot\xi_t^{+}=0\) if \(\dot\xi_t\leq 0\).

It follows from (3) that, for \((u,|\Delta|)\to\infty\), the limiting distributions of \(\eta_u(\Delta)\) and \(\widetilde\eta_u(\Delta)\) coincide.

Consider the \(k\)-fold integrals defined by the formulas

\[ J_{(k)}(\Delta;u,\tau)= \int_{\substack{t_i\in\Delta,\ |t_i-t_j|>\tau\\ i,j=1,\ldots,k}} \cdots \int \mathbf M\left(\prod_{i=1}^{k}\dot\xi_{t_i}^{+}\,\middle|\,\xi_{t_j}=u,\ j=1,\ldots,k\right) p_{t_1\ldots t_k}(u\ldots u)\,dt_1\ldots dt_k . \tag{4} \]

It is easy to show that under the assumptions made \(J_{(k)}(\Delta; u,\tau)<\infty\).
For the proof of the main assertions the following lemma is useful.

Lemma 1. Under a coordinated increase
\[ (u,|\Delta|)\to\infty,\qquad |\Delta|=\frac{2\pi\mu}{\sqrt{-\rho''(0)}}e^{u^2/2} \]
there exist, for every \(k\), numbers \(\tau_k>0\) such that for all \(\tau\), \(\tau_k<\tau<\sqrt{|\Delta|}\),
\[ \lim_{(u,|\Delta|)\to\infty} J_{(k)}(\Delta;u,\tau)=\mu^k,\qquad k=1,2,\ldots . \tag{5} \]

Let us now consider the behavior of the factorial moments \(\widetilde J_k(\Delta,u)\) for the random variables \(\widetilde\eta_u(\Delta)\); recall that
\[ \widetilde J_{(k)}(\Delta,u)=\mathbf M\widetilde\eta_u(\Delta)\,[\widetilde\eta_u(\Delta)-1]\cdots[\widetilde\eta_u(\Delta)-k+1]. \]

Theorem 2. Under a coordinated increase \((u,|\Delta|)\to\infty\),
\[ |\Delta|=\frac{2\pi\mu}{\sqrt{-\rho''(0)}}e^{u^2/2}, \]
and assuming conditions (1), (2) are satisfied,
\[ \lim_{(u,|\Delta|)\to\infty}\widetilde J_{(k)}(\Delta,u)=\mu^k,\qquad k=1,2,\ldots . \tag{6} \]

In the proof of Theorem 2, assertion (5) is used, as well as the fact that the limiting behavior of \(\widetilde J_{(k)}\) and \(J_{(k)}\) is the same. Here the method used in the proof of Theorem 1 of the author’s paper \((^1)\) proves useful. The main result of the paper is obtained as a consequence of the theorem on convergence of moments (\((^2)\), p. 198).

Theorem 3. If, for the correlation function \(\rho(t)\) of a stationary Gaussian process, (1), (2) are satisfied, then under a coordinated increase \((u,|\Delta|)\to\infty\),
\[ \lim_{(u,|\Delta|)\to\infty}\mathbf P\{\eta_u(\Delta)=k\} =\frac{(\mu|\Delta|)^k}{k!}e^{-\mu|\Delta|},\qquad k=0,1,\ldots . \]

It can be shown that the joint multidimensional distributions will also be Poisson. Theorem 3 is a generalization of Cramér’s result \((^3)\), in which the existence of four derivatives of the correlation function \(\rho(t)\) and \(\rho(t)=o(t^{-\varepsilon})\), \(t\to\infty\), \(\varepsilon>0\), are assumed. Using Theorem 3, one can obtain a generalization of one more result of Cramér \((^4)\).

Theorem 4. If conditions (1), (2) are satisfied, then
\[ \lim_{T\to\infty}\mathbf P\left\{\max_{0\le t\le T}\xi_t\le \sqrt{2\ln T}+\frac{z-A}{\sqrt{2\ln T}}\right\} =e^{-e^{-z}},\qquad A=\ln\frac{2\pi}{\sqrt{-\rho''(0)}}. \]

Moscow State University
named after M. V. Lomonosov

Received
19 V 1966

REFERENCES

1. Yu. K. Belyaev, Theory of Probability and Its Applications, 11, 1, 120 (1966).
2. M. Loeve, Probability Theory, IL, 1962.
3. H. Cramér, Ark. Matem., 6, 20, 337 (1966).
4. H. Cramér, Theory of Probability and Its Applications, 10, 1, 137 (1965).