Abstract Generated abstract
This paper studies conditions for the existence of factorial moments of the number of upward crossings of a fixed level by a Gaussian stationary process. Starting from the standard integral representation for these moments, it gives sufficient conditions in terms of integrability of conditional mean and variance expressions, and proves the necessity of one such condition using auxiliary inequalities. Under assumptions on the existence of the kth mean-square derivative and local behavior of the correlation function near the origin, the paper derives asymptotics of the relevant covariance determinants and shows that the mth moment is finite exactly when m is at most k squared plus 2k. As a corollary, for stationary Gaussian processes with rational spectral density and exactly k mean-square derivatives, the highest finite crossing moment has this order.
Full Text
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.