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UDC 519.217
MATHEMATICS
V. I. PITERBARG
ON THE EXISTENCE OF MOMENTS OF THE NUMBER OF LEVEL CROSSINGS BY A GAUSSIAN STATIONARY PROCESS
(Presented by Academician A. N. Kolmogorov on 5 I 1968)
It is known \((^{1,2})\) that the \(m\)-th factorial moment of the number of crossings from below upward of the level \(a\) by a Gaussian process without linear relations \(\xi_t\) on the interval \(\Delta\) is equal to
\[ \begin{aligned} d_m(\Delta,a) &= \int_{\{t_i\in\Delta,\; t_i\ne t_j,\; i,j=1,\ldots,m\}} M\left\{\prod_{i=1}^{m}\dot{\xi}_{t_i}^{+}\mid \xi_{t_j}=a,\; j=1,\ldots,m\right\} \\ &\qquad\qquad\times P_{t_1\ldots t_m}(a,\ldots,a)\,dt_1\ldots dt_m, \end{aligned} \tag{1} \]
where \(P_{t_1\ldots t_m}(x_1,\ldots,x_m)\) is the density of the joint distribution of the quantities \(\xi_{t_1},\ldots,\xi_{t_m}\) with covariance matrix \(R_{11}(t_1,\ldots,t_m)\),
\[ \dot{\xi}_{t_i}=d\xi_{t_i}/dt_i,\qquad \dot{\xi}_{t_i}^{+}=\tfrac12(\dot{\xi}_{t_i}+|\dot{\xi}_{t_i}|). \]
Below we study the question of conditions for the existence of the factorial moments specified by the integrals (1). Denote
\[ I(l)= \int_{\{t_i\in\Delta,\; t_i\ne t_j,\; i,j=1,\ldots,m\}} \frac{\mu_{i_1}\ldots\mu_{i_l}\sigma_{i_{l+1}}\ldots\sigma_{i_m}} {\sqrt{|R_{11}(t_1,\ldots,t_m)|}} \,dt_1,\ldots,dt_m, \]
where
\[ \mu_i=M\{\dot{\xi}_{t_i}\mid \xi_{t_j}=a,\; j=1,\ldots,m\},\qquad \sigma_i^2=D\{\dot{\xi}_{t_i}\mid \xi_{t_j}=a,\; j=1,\ldots,m\}, \]
\((i_1,\ldots,i_m)\) is a permutation of the indices \((1,\ldots,m)\).
Theorem 1. For the existence of the \(m\)-th moment of the number of crossings of the level \(a\) by a Gaussian process it is sufficient that the integrals \(I(l)\) \((l=0,\ldots,m)\) exist, and necessary that the integral \(I(0)\) exist.
Sufficiency was proved by Yu. K. Belyaev \((^{1})\), Lemma 4. Necessity follows from two lemmas.
Lemma 1. For any vector \(a\) and nondegenerate matrix \(A\),
\[ a'A^{-1}a=\min_x(x'Ax-2a'x). \tag{2} \]
Lemma 2. There exist \(\varepsilon>0\) and \(0<\alpha<1\) such that, if \(\max_{i,j}|t_i-t_j|<\varepsilon\), then
\[ M\left\{\prod_{i=1}^{m}\dot{\xi}_{t_i}^{+}\mid \xi_{t_j}=a,\; j=1,\ldots,m\right\} \ge \alpha\sigma_1\cdots\sigma_m. \]
Theorem 2. Suppose:
-
The \(k\)-th mean-square derivative of the stationary Gaussian process exists.
-
In some interval \((0,\delta)\) there exist and are continuous \(\rho^{(2k+1)}(t)\) and \(\rho^{(2k+2)}(t)\), respectively the \((2k+1)\)-st and \((2k+2)\)-nd derivatives of the correlation function of \(\xi_t\).
- There exist limits
\[ \lim_{t \downarrow 0} \frac{\rho^{(2k)}(0)-\rho^{(2k)}(t)}{t} = \rho_{+}^{(2k+1)}(0) \ne 0, \]
\[ \lim_{t \downarrow 0} \frac{\rho_{+}^{(2k+1)}(0)-\rho^{(2k+1)}(t)}{t} = \rho_{+}^{(2k+2)}(0). \]
Under these conditions \(\alpha_{(m)}(\Delta,a)<\infty\) if and only if \(m\le k^2+2k\).
Proof. We find the asymptotics, as \(t_i-t_j\to 0\), of the functions
\[ \frac{\mu_{i_1}\cdots \mu_{i_l}\sigma_{i_{l+1}}\cdots \sigma_{i_m}} {\sqrt{|R_{11}(t_1,\ldots,t_m)|}}. \tag{3} \]
We shall carry out the estimate when all \(t_i-t_j\to 0\). In the remaining cases the estimate is analogous. At the same time, for brevity, by \(\lim\) we mean passage to the limit as \(\max |t_i-t_j|\to 0\).
Consider the quantities
\[ \zeta_{t_i t_{i+1}}^{(k)} = \frac{\xi_{t_{i+1}}^{(k)}-\xi_{t_i}^{(k)}}{\sqrt{|t_{i+1}-t_i|}}. \]
It is easy to see that
\[ \lim M\zeta_{t_i t_{i+1}}^{(k)}\zeta_{t_l t_{l+1}}^{(k)} = \begin{cases} 0, & \text{if } i\ne l,\\ 2\rho_{+}^{(2k+1)}(0), & \text{if } i=l. \end{cases} \]
Denote, following (1), by \(L_{t_1\ldots t_{k+1}}[\xi_t]=[\xi_{t_1},\ldots,\xi_{t_k}]/k!\), where \([\xi_{t_1},\ldots,\xi_{t_k}]\) is the \(k\)-th divided difference of \(\xi_t\).
The quantities
\[ \zeta_{t_1\ldots t_{k+2}} = \frac{L_{t_1\ldots t_{k+1}}[\xi_t]-L_{t_2\ldots t_{k+2}}[\xi_t]} {\sqrt{|t_1-t_{k+2}|}} \]
possess the same properties as the \(\zeta_{t_i t_{i+1}}^{(k)}\) introduced above.
It is known that
\[ \mu_i=\frac{|R_{\mu_i}|}{|R_{11}|} = \left| \begin{array}{c:c} R_{11} & \dfrac{\partial \rho(t_i-t_j)}{\partial t_i}\\ \cdots & \cdots\\ \hdashline a\ \cdots\ a & 0 \end{array} \right| \,|R_{11}(t_1,\ldots,t_m)|^{-1}, \]
\[ \sigma_i^2=\frac{|R_{\sigma_i}|}{|R_{11}|} = \left| \begin{array}{c:c} R_{11} & \dfrac{\partial \rho(t_i-t_j)}{\partial t_i}\\ \cdots & \cdots\\ \hdashline \cdots\ \dfrac{\partial \rho(t_i-t_j)}{\partial t_i}\ \cdots & \dfrac{\partial^2\rho(0)}{\partial t^2} \end{array} \right| \,|R_{11}(t_1,\ldots,t_m)|^{-1}. \]
We perform the following transformations on \(|R_{11}|\). At the first step, from each \(i\)-th column we subtract the \((i-1)\)-st and divide by \(t_i-t_{i-1}\), and from each \(i\)-th row we subtract the \((i-1)\)-st and divide by \(t_i-t_{i-1}\).
At the second step we do the same as at the first, without touching the first row and column, and dividing by \(t_i-t_{i-2}\), and so on up to the \(k\)-th step (see (1)). At the \((k+1)\)-st step, from the \(i\)-th column (\(i=m,\ldots,k+2\)) we subtract the \((i-1)\)-st and divide by \(\sqrt{|t_i-t_{i-k-1}|}\); we do the same with the rows and pass to the limit.
Having carried out analogous transformations on \(R_{\sigma_i}\) and \(R_{\mu_i}\), we obtain
asymptotic formulas.
\[ \frac{1}{2^{m-k-1}}\lim \frac{|R_{11}(t_1,\ldots,t_m)|} {\prod_{l=1,\ldots,k}\prod_{i=l+1,\ldots,m}(t_i-t_{i-l})^2 \prod_{i=k+2,\ldots,m}|t_i-t_{i-k-1}|} = |M\eta\eta'|\,[\rho_+^{(2k+1)}(0)]^{m-k-1}, \]
\[ \lim \frac{|R_{\sigma_i}(t_1,\ldots,t_m)|} {|R_{11}(t_1,\ldots,t_m)| \prod_{l=1,\ldots,k-1}(t_i-t_{i-l})|t_i-t_{i-k}|} = 2\rho_+^{(2k+1)}(0), \]
\[ \lim \frac{|R_{\mu_i}(t_1,\ldots,t_m)|} {|R_{11}(t_1,\ldots,t_m)| \prod_{l=1,\ldots,k-1}(t_i-t_{i-l})\sqrt{|t_i-t_{i-k}|}} = k\rho_+^{(2k+1)}(0), \]
where \(\eta=(\xi_{t_0},\ldots,\xi_{t_0}^{(k)})\).
On the basis of these formulas we conclude that, for \(m=k^2+2k\), the singularities of the function (3) are integrable, whereas already for \(m=k^2+2k+1\) \(I(0)\) does not exist.
Corollary. For Gaussian stationary processes with rational spectral density that have exactly \(k\) derivatives in the mean-square sense, the last finite moment of the number of level crossings has order \(k^2+2k\).
The author expresses his gratitude to Yu. K. Belyaev for supervising the work.
Moscow State University
named after M. V. Lomonosov
Received
27 XII 1967
REFERENCES
- Yu. K. Belyaev, Theory of Probability and Its Applications, 11, 1, 120 (1966).
- H. Cramér, M. R. Leadbetter, Ann. Math. Statist., 36, 6, 1656 (1965).
- A. O. Gelfond, The Calculus of Finite Differences, “Nauka,” Moscow, 1967.
- E. Beckenbach, R. Bellman, Inequalities, Moscow, 1965.