Abstract
Full Text
UDC 519.21
MATHEMATICS
M. P. ERSHOV
A LIMIT THEOREM FOR THE NUMBER OF LEVEL CROSSINGS BY A STATIONARY GAUSSIAN PROCESS
(Presented by Academician Yu. V. Linnik on 28 II 1967)
Consider a continuous with probability 1 stationary Gaussian process \(\xi_t\) with correlation function \(r_t\). Let \(\xi_t\) be differentiable in mean square (m.s.), and let \(N(s,t)\) denote the number of crossings from below to above of the level \(u\) by the process \(\xi_t\) on \([s,t]\). From \((^1)\) it follows that \(E N(s,t)<\infty\).
Lemma 1. If \(r_t \to 0\) as \(t \to \infty\), then
\[ D N(0,T)\xrightarrow[T\to\infty]{}\infty \]
for all \(u\).
Proof. Let \(D N(0,T)<\infty\), otherwise the assertion of the lemma is trivial. Put
\[ X_k(\tau)=N(k\tau,(k+1)\tau)-E N(k\tau,(k+1)\tau),\qquad k=0,1,\ldots \tag{1} \]
For any \(\tau\), \(\{X_k(\tau)\}\) is a stationary sequence, and, by Theorem 2 on p. 119 in \((^2)\), \(E X_0 X_k \to 0\) as \(k\to\infty\). Choose \(\tau\) so that
\[ E N(0,\tau)=1/2. \tag{2} \]
Let \(\underline{\lim}_{\,n}\,D N(0,n\tau)<\infty\). Then, by Theorem 18.2.2 in \((^3)\), there is a representation:
\[ X_k=Y_{k+1}-Y_k, \]
where \(Y_k=U^kY_0,\; Y_0\in H(X)\), and \(H(X)\) is the closure, in the m.s. sense, of the linear span of the set \(\{X_0,X_1,\ldots\}\), while \(U\) is a unitary operator in \(H(X)\) defining \(X_k\): \(X_k=U^kX_0\). From (2) it follows that
\[ \mathbf P\{|Y_{2n+1}-Y_0|<1/2\}=0. \tag{3} \]
Let \(I=(a,a+1/2)\) be an interval such that \(\mathbf P\{Y_0\in I\}=p\ne0\). The variables \(Y_n\), obviously, are measurable with respect to \(\xi_t,\;0\le t<\infty\); therefore, by the mentioned Theorem 2 in \((^2)\),
\[ \mathbf P\{Y_0,Y_n\in I\}\xrightarrow[n\to\infty]{}p^2, \]
which contradicts (3). Thus, \(\underline{\lim}_{\,n}\,D N(0,n\tau)=\infty\). Let \(T\to\infty\), and put \(n=[T/\tau]\). We have
\[ D N(0,T)>D N(0,n\tau)-2\sqrt{D N(0,n\tau)\,E\bigl(N(0,\tau)\bigr)^2}\to\infty. \]
Lemma 2. If \(\xi_t\) is twice differentiable in the m.s. sense and
\[ \int_0^\infty t\bigl(|r_t|+|r_t'|+|r_t''|\bigr)\,dt<\infty, \]
then, as \(T\to\infty\),
\[ D N(0,T)=\sigma^2 T(1+o(1)) \]
and \(\sigma\ne0\).
Proof. From \((^5)\) it follows that \(D N(0,T)<\infty\). Let \(\tau\) be fixed.
\[ D N(0,n\tau)=n E X_0^2(\tau)+2\sum_{k=1}^{n-1}(n-k)E X_0(\tau)X_k(\tau), \tag{4} \]
where \(X_k(\tau)\) is defined from (1). In the same way as in (4) when finding the factorial moments of \(N(0,T)\), one can show that
\[ \mathbf{E}X_0(\tau)X_k(\tau)= \int_0^\tau\left(\int_{k\tau}^{(k+1)\tau}(F_{t-s}-F_\infty)\,dt\right)ds,\qquad k>0, \tag{5} \]
where
\[ F_{t-s}=\int_0^\infty\int_0^\infty y_1y_2\varphi_{t-s}(u,u,y_1,y_2)\,dy_1dy_2,\qquad F_\infty=\lim_{t\to\infty}F_t, \]
and \(\varphi_{t-s}(x_1,x_2,y_1,y_2)\) is the density of the vector \(\{\xi_s,\xi_t,\dot\xi_s,\dot\xi_t\}\). (We note that, by the conditions of the lemma, \(r_t,r_t',r_t''\to0\) as \(t\to\infty\), so the definition of \(F_\infty\) makes sense.) Let \(R_{t-s}\) and \(R_\infty\) be the correlation matrices corresponding to \(\varphi_{t-s}\) and \(\varphi_\infty=\lim_{t\to\infty}\varphi_t\). Put \(\delta_{t-s}=\|R_{t-s}-R_\infty\|\), where \(\|(a_{ij})\|=\max_{ij}|a_{ij}|\). It is easy to obtain the estimate
\[ \begin{aligned} &\left|\varphi_{t-s}(u,u,y_1,y_2)-\varphi_\infty(u,u,y_1,y_2)\right|\\ &\quad < C\exp\left\{-(4r_0-4r_0'')^{-1}(y_1^2+y_2^2)\delta_{t-s}|R_{t-s}|^{-3/2}(1+y_1^2+y_2^2)\right\}, \end{aligned} \tag{6} \]
where \(C\) depends only on \(u,r_0,r_0''\). From (5) and (6) we obtain
\[ \sum_1^\infty |\mathbf{E}X_0X_k| <2\int_0^\infty t\,|F_t-F_\infty|\,dt< \]
\[ <2a\int_0^a |F_t-F_\infty|\,dt +2C_1\int_a^\infty t\delta_t |R_t|^{-3/2}\,dt. \tag{7} \]
The first integral is finite, since \(DN(0,a)<\infty\). It is obvious that \(|R_t|\) is bounded below uniformly in \(t\) on \([a,\infty)\) for any \(a>0\). In addition, \(\delta_t<|r_t|+|r_t'|+|r_t''|\); therefore the second integral on the right-hand side of (7) is also finite. Finally we obtain
\[ \left|\sum_1^\infty \mathbf{E}X_0(\tau)X_k(\tau)\right|<K, \]
where \(K\) does not depend on \(\tau\). By Lemma 1, \(\mathbf{E}X_0^2(\tau)\to\infty\) as \(\tau\to\infty\). Choose \(\tau\) so that \(\mathbf{E}X_0^2(\tau)>2K\). Now (4) can be rewritten as follows:
\[ DN(0,n\tau)= n\left(\mathbf{E}X_0^2(\tau)+2\sum_1^\infty \mathbf{E}X_0(\tau)X_k(\tau)+o(1)\right) = \]
\[ =n\sigma_1^2(1+o(1)),\qquad \sigma_1\ne0. \]
Put \(\sigma^2=\sigma_1^2/\tau\), and let \(T\to\infty\), \(n=[T/\tau]\).
\[ \left|\frac{DN(0,T)}{\sigma^2T}-1\right| < \frac{|DN(0,T)-DN(0,n\tau)|}{\sigma^2T} + \]
\[ +DN(0,n\tau)\left(\frac{1}{\sigma^2n\tau}-\frac{1}{\sigma^2T}\right) +\left|\frac{DN(0,n\tau)}{\sigma^2n\tau}-1\right|=o(1). \]
Lemma 3. Let \(R_t\) denote the correlation matrix of the vector \(\{\xi_0,\xi_t,\dot\xi_0,\dot\xi_t\}\). If \(r_t^{(8)}\) is continuous, then in some neighborhood of zero
\[ |R_t|=ct^8+o(t^8),\qquad c>0. \]
Proof. Putting \(r_0=1\), we find
\[ |R_t|=(r_t^2+r_0''-r_tr_t'')^2-(r_t'-r_tr_0'')^2. \]
It is enough to expand the terms in the parentheses up to \(t^6\), since the expansion will begin with \(t^2\). We obtain
\[ |R_t|=\frac{t^8}{144}\left(r_0^{(4)}-r_0''^{\,2}\right)\left(r_0^{(6)}r_0''-r_0^{(4)\,2}\right)+o(t^8). \]
From the positive definiteness of \(R_t\) it follows that the coefficient of \(t^8\) is nonnegative. Let \(r_0^{(4)}=r_0''/2\). This means that \(\mathrm E\xi_0^2\mathrm E\dot\xi_0^2=(\mathrm E\xi_0\dot\xi_0)^2\). But in the range space of \(\xi_t\), which is the closure in mean square of the linear span of the variables \(\xi_{t_k}\), such an equality is possible only under the condition \(\mathrm E(\dot\xi_0-\lambda \xi_0)^2=0\), where \(\lambda\) is some number; this cannot occur when the process \(\xi_t\) is Gaussian. Similarly we verify that \(r_0^{(6)}r_0''-r_0^{(4)2}>0\).
Lemma 4. If there exists and is continuous in mean square \(d^4\xi_t/dt^4\), and \(r_t,r_t',r_t''=O(t^{-\alpha})\) as \(t\to\infty\) for some \(\alpha>13\), then for all \(u\)
\[ \mu_4(T)=\mathrm E\bigl(N(0,T)-\mathrm EN(0,T)\bigr)^4=O(T^2)\quad\text{as }T\to\infty . \]
Proof. From (5) it follows that \(\mu_4(T)<\infty\). In the notation (1) we have
\[ \begin{aligned} \mu_4(n\tau)&< n\mathrm EX_0^2+\sum_{i\ne j}\left|\mathrm E\left(X_i^2X_j^2+X_i^3X_j\right)\right| +\sum_{i\ne j\ne k}\left|\mathrm EX_i^2X_jX_k\right|\\ &\quad+\sum_{i\ne j\ne k\ne l}\left|\mathrm EX_iX_jX_kX_l\right|\\ &=O\left(n^2+\sum_{i<j-1<k-2}\left|\mathrm EX_i^2X_jX_k\right| +\sum_{i<j-1<k-2<l-3}\left|\mathrm EX_iX_jX_kX_l\right|\right). \end{aligned} \tag{8} \]
Put \(N(i\tau,(i+1)\tau)=N_i,\; N_i(N_i-1)=N_i^{(2)}\). Consider the second term on the right-hand side of (8):
\[ \mathrm EX_i^2X_jX_k=\mathrm E\bigl(N_i^{(2)}N_j+N_iN_j-(N_i^{(2)}+N_i)\mathrm EN_j-(\mathrm EN_i)^2N_j\bigr)X_k . \tag{9} \]
As in (4), we obtain:
\[ \mathrm EN_i^{(2)}N_jX_k = \int_{\Delta_i\times\Delta_i}dt_1dt_2 \int_{\Delta_j}dt_3 \int_{\Delta_k}\bigl(F_{t_1t_2t_3t_4}-F_{t_1t_2t_3\infty}\bigr)\,dt_4, \tag{10} \]
where
\[ F_{t_1t_2t_3t_4} = \int_0^\infty\cdots\int_0^\infty y_1\cdots y_4 \varphi_{t_1\ldots t_4}(u,\ldots,u,y_1,\ldots,y_4)\,dy_1\cdots dy_4 \]
and \(\varphi_{t_1\ldots t_4}(x_1,\ldots,x_4,y_1,\ldots,y_4)\) is the density of the vector
\(\{\xi_{t_1},\ldots,\xi_{t_4},\dot\xi_{t_1},\ldots,\dot\xi_{t_4}\}\),
\(F_{t_1t_2t_3\infty}=\lim_{t_4\to\infty}F_{t_1t_2t_3t_4}\), with \(t_1\ne t_2\in\Delta_i=(i\tau,(i+1)\tau)\), \(t_3\in\Delta_j\). Denote by \(R_{t_2-t_1}\) the correlation matrix of the vector \(\{\xi_{t_1},\xi_{t_2},\dot\xi_{t_1},\dot\xi_{t_2}\}\). By Lemma 3 there exists \(t_0\) such that for all \(t_1,t_2\) for which \(|t_1-t_2|<t_0\),
\[ |R_{t_2-t_1}|>c(t_2-t_1)^8,\qquad c>0. \]
Introduce the function
\[ \beta_k=\beta_k(\tau)=\max_{s>k\tau}\bigl(|r_s|+|r_s'|+|r_s''|\bigr). \]
Put
\[ A_0=\{t_1,t_2\in\Delta_i:\ |t_1-t_2|<\beta_{j-i-1}^{1/13}+\beta_{k-j-1}^{1/13}\}, \]
\[ A_1=\{t_1,t_2\in\Delta_i:\ |t_1-t_2|\ge t_0\},\qquad A_2=(\Delta_i\times\Delta_i)\setminus(A_0\cup A_1). \]
For sufficiently large \(\tau\), \(A_0A_1=\varnothing\) for all \(i<j-1<k-2\). We split the outer integral in (10) into three, over the sets \(A_0,A_1,A_2\), respectively. The first integral will be of order
\(\beta_{j-i-1}^{1/13}+\beta_{k-j-1}^{1/13}\), as follows from the proof of Theorem 3 in (5). The second is estimated in the same way as in the proof of Lemma 2. To estimate the integral over \(A_2\), we use the obvious analogue of relation (6). We obtain:
\[ \left|F_{t_1\ldots t_4}-F_{t_1t_2t_3\infty}\right| < C\delta_{t_1\ldots t_4} \left[\min\{|R_{t_1\ldots t_4}|,\ |R_{t_1t_2t_3\infty}|\}\right]^{-3/2}, \]
where \(C\) depends only on \(u,r_0,r_0'\). Obviously, on \(A_2\)
\[ \min\{|R_{t_1\ldots t_4}|,\ |R_{t_1\ldots\infty}|\} > c_1(\beta_{j-i-1}^{1/13}+\beta_{k-j-1}^{1/13}), \]
and, since \(\delta_{t_1\ldots t_4}=|R_{t_1\ldots t_4}-R_{t_1\ldots\infty}|<\beta_{k-j-1}\), it follows that
\[ \left|F_{t_1\ldots t_4}-F_{t_1\ldots\infty}\right| < C_1(\beta_{j-i-1}^{1/13}+\beta_{k-j-1}^{1/13}). \]
Thus, for the integral (10) there is the estimate
\[ \left|\mathrm EN_i^{(2)}N_jX_k\right| < C_2(\beta_{j-i-1}^{1/13}+\beta_{k-j-1}^{1/13}). \]
Similarly one estimates \(|EN_i^{(2)}X_k|\). For the other terms in (10) we easily obtain
\(|EN_iN_jX_k|<C_3\beta_{k-j-1}\), \(|EN_j^2X_k|<C_4\beta_{k-j-1}\). Finally,
\[ \sum_{i<j-1<k-2} |EX_i^2X_jX_k| = O\left( \sum_{i<j-1<k-2} \left(\beta_{j-i-1}^{1/3}+\beta_{k-j-1}^{1/3}\right)+n^2 \right) = O(n^2). \]
For the last term in (8) we find the estimate:
\[ \sum_{i<j-1<k-2<l-3} |EX_iX_jX_kX_l| = \]
\[ = O\left( \sum_{i<j-1<k-2<l-3} \min(\beta_{j-i-1},\beta_{l-k-1}) \right) = O\left(n^2\sum_0^\infty i\beta_i\right) = O(n^2). \]
Thus, \(\mu_4(n\tau)=O(n^2)\) for some sufficiently large \(\tau\). Hence the assertion of Lemma 4 follows.
Suppose now that \(\xi_t\) satisfies the strong mixing condition in the sense that
\[ \rho(t)= \sup_{\substack{A\in\mathcal F^0_{-\infty}\\ B\in\mathcal F^\infty_t}} \left|\mathbf P(AB)-\mathbf P(A)\mathbf P(B)\right| \underset{t\to\infty}{\longrightarrow}0, \tag{11} \]
where \(\mathcal F_a^b\) is the \(\sigma\)-algebra generated by \(\xi_t,\ t\in[a,b]\). As a consequence of Lemmas 2 and 4, the following is obtained.
Theorem. Under the conditions of Lemma 4 and (11)
\[ \mathbf P\left\{ \frac{N(0,T)-EN(0,T)}{\sqrt{DN(0,T)}}<z \right\} \to \frac1{\sqrt{2\pi}}\int_{-\infty}^{z} e^{-x^2/2}\,dx \tag{12} \]
as \(T\to\infty\), for all \(u\).
Proof. According to Theorem 18.4.1 in (3), in order to prove (12) for \(T=n\tau\), where \(\tau\) is fixed, it is sufficient to verify, for any integer-valued function \(p=p(n)\) such that \(p\to\infty\), \(p=o(n)\), and any \(\varepsilon>0\), the relation
\[ \frac{n}{p\sigma_n^2} \int_{|z|>\varepsilon\sigma_n} z^2\,dF_p(z)\to0 \quad\text{as } n\to\infty, \]
where \(\sigma_n^2=DN(0,n\tau)\), \(F_p(z)=\mathbf P\{N(0,p\tau)-EN(0,p\tau)<z\}\). We have
\[ \frac{n}{p\sigma_n^2} \int_{|z|>\varepsilon\sigma_n} z^2\,dF_p(z) < \frac{n\mu_4(p\tau)}{\varepsilon^2p\sigma_n^4} = O\left(\frac pn\right) = o(1). \]
Now, for arbitrary \(T\), (12) is proved with the aid of the obvious proposition: if \(\mathbf P\{\xi_n+\eta_n<x\}\to F(x)\), where \(F\) is some distribution function and \(\eta_n\to0\) in probability, then \(\mathbf P\{\xi_n<x\}\to F(x)\).
I take this opportunity to express my gratitude to Prof. O. V. Sarmanov for proposing the problem and for his attention to its solution.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
25 II 1967
REFERENCES
- K. Ito, J. Math. Kyoto Univ., 3, 2, 207 (1964).
- K. Ito, Probability Processes, vol. 1, IL, 1960.
- I. A. Ibragimov, Yu. V. Linnik, Independent and Stationarily Related Random Variables, “Nauka,” 1965.
- H. Cramér, M. R. Leadbetter, Ann. Math. Stat., 36, 6, 1656 (1965).
- Yu. K. Belyaev, Theory of Probability and Its Applications, 11, 1, 120 (1966).