Abstract
Full Text
UDC 519.21
MATHEMATICS
P. I. YUDITSKAYA
ON THE ASYMPTOTIC BEHAVIOR OF THE MAXIMUM OF GAUSSIAN RANDOM FIELDS
(Presented by Academician Yu. V. Linnik on 20 II 1970)
Let \(\xi(t)\) be a stationary separable Gaussian real random process with spectral density \(f(\lambda)\), satisfying the condition
\[ \int_{0}^{\infty} \lambda^2 [\ln(1+\lambda)]^a f(\lambda)\,d\lambda < \infty \]
for some \(a>1\).
T. Cramer \((^1)\) proved that
\[ \lim_{T\to\infty} P\left\{\left|\max_{0\leq t\leq T}\xi(t)-\sqrt{2\ln T}\right|<\frac{\ln\ln T}{\sqrt{2\ln T}}\right\}=1. \]
M. G. Shur \((^2)\) established that, under the same assumptions, with probability one
\[ \max_{0\leq t\leq T}\xi(t)-\sqrt{2\ln T}\longrightarrow 0. \]
In the present note the asymptotic behavior of the maximum of random fields is investigated. The methods of proof differ from \((^1,^2)\), since in the case of functions of several variables one cannot use estimates connected with the number of level crossings.
Consider a separable real Gaussian homogeneous isotropic random field \(\xi(\bar{x})\) in \(n\)-dimensional Euclidean space \(R^n\). As is known,
\[ M\xi(\bar{x}_1)\xi(\bar{x}_2)=R_1(\bar{x}_1-\bar{x}_2)=R_2(r), \]
where \(r^2=(\bar{x}_1-\bar{x}_2)(\bar{x}_1-\bar{x}_2)'\). Without loss of generality, we shall assume that \(M\xi(\bar{x})=0,\ D\xi(\bar{x})=1\).
Let \(\nu\) be Lebesgue measure defined on the \(\sigma\)-algebra \(\mathfrak{M}\) of Lebesgue-measurable sets in \(R^n\). Consider in \(R^n\) the parallelepiped \(D(X_1,\ldots,X_n)=(0\leq x_1\leq X_1,\ldots,0\leq x_n\leq X_n)\) and a measurable closed simply connected domain \(E_1\), bounded by the surface \(\Phi_1\), \(\nu(E_1)=1,\ \bar{0}\in E_1\).
Make a similarity transformation of the surface \(\Phi_1\) with center at the origin and similarity coefficient \(K>1\); we obtain the surface \(\Phi_k\), bounding the domain \(E_k\).
Theorem. Suppose that, with probability one, there exist continuous first and second partial derivatives of the random field \(\xi(\bar{x})\), and
\[ P\{\det \|\partial^2 \xi(\bar{x})/\partial x_i\partial x_j\|=0\}=0. \]
Let the correlation function of the field satisfy the conditions:
\[ \text{I.}\quad \left| \frac{\partial^4 R_1(\bar{x})}{\partial x_1^{\varepsilon_1}\cdots \partial x_n^{\varepsilon_n}} - \frac{\partial^4 R_1(\bar{0})}{\partial x_1^{\varepsilon_1}\cdots \partial x_n^{\varepsilon_n}} \right| < N_1\sum_{i=1}^n |x_i|^\delta, \]
\[ \varepsilon_i=0,2,4;\quad N_1=\mathrm{const}>0;\quad \sum_{i=1}^n \varepsilon_i=4;\quad \delta>0. \]
\[ \text{II.}\quad |R_2(r)|<N_2/r^n,\quad N_2=\mathrm{const}. \]
Then for every \(\varepsilon>0\), almost surely there exist random \(K^0(\varepsilon)<\infty,\ X_i^0(\varepsilon)<\infty\ (i=1,\ldots,n)\), such that for \(K>K^0(\varepsilon)\)
\[ \left|\max_{\bar{x}\in E_k}\xi(\bar{x})-\sqrt{2\ln\nu(E_k)}\right| < (2+\varepsilon)\frac{\ln\ln\nu(E_k)}{\sqrt{2\ln\nu(E_k)}}, \]
and for \(X_i>X_i^0(\varepsilon)\), \(i=1,\ldots,n\),
\[ \left| \max_{\bar x\in D(X_1,\ldots,X_n)} \xi(\bar x) -\left(2\ln\prod_{i=1}^n X_i\right)^{1/2} \right| < (n+1+\varepsilon)\ln\ln\prod_{i=1}^n X_i \Big/ \left(2\ln\prod_{i=1}^n X_i\right)^{1/2}. \]
From the work of Yu. K. Belyaev \((^3)\) it follows that, under the conditions of the theorem, there exists \(\mu_c\), the intensity of the mean number of local maxima of the random field exceeding the level \(c\).
Lemma 1. For arbitrary \(c>0\),
\[ \mu_c \leq f(n)e^{-c^2/2}, \]
where \(f(n)\) is finite for every \(n\) and does not depend on \(c\).
Lemma 2. For every \(c>0\), \(k>1\),
\[ P\left\{\max_{\bar x\in E_k}\xi(\bar x)>c\right\} \leq N_3\nu(E_k)e^{-c^2/2}. \tag{1} \]
Moreover, if \(X_i>1\) \((i=1,\ldots,n)\), then
\[ P\left\{ \max_{\bar x\in D(X_1,\ldots,X_n)} \xi(\bar x)>c \right\} \leq N_3\prod_{i=1}^n X_i e^{-c^2/2}. \tag{2} \]
Here \(N_3\) is a constant that does not depend on \(c,k,X_1,\ldots,X_n\).
Lemma 3. Under the conditions of the theorem, for every \(c>0\),
\[ P\left\{\max_{\bar x\in G}\xi(\bar x)<c\right\} \leq N_4c^2\left(e^{c^2/2}+e^{c^2/3}\ln\nu(G)\right)/\nu(G), \tag{3} \]
where \(G\) coincides with \(D(X_1,\ldots,X_n)\) or with \(E_k\); \(N_4\) does not depend on \(c,X_1,\ldots,X_n,k\), and \(X_i>1\) \((i=1,\ldots,n)\), \(k>1\).
Lemma 4. Suppose the following series converge:
\[ \sum_{m_1=1}^{\infty}\cdots\sum_{m_n=1}^{m} P\left\{ \max_{\bar x\in D(e^{m_1},\ldots,e^{m_n})}\xi(\bar x)> \right. \]
\[ \left. > \left[ 2\sum_{i=1}^n m_i + \left(\ln\sum_{i=1}^n m_i\right)(n+1+\varepsilon) \right] \Big/ \left(2\sum_{i=1}^n m_i\right)^{1/2} \right\}, \]
\[ \sum_{m_1=1}^{\infty}\cdots\sum_{m_n=1}^{\infty} P\left\{ \max_{\bar x\in D(e^{m_1},\ldots,e^{m_n})}\xi(\bar x)< \right. \]
\[ \left. < \left[ 2\sum_{i=1}^n m_i - \left(\ln\sum_{i=1}^n m_i\right)(n+1+\varepsilon) \right] \Big/ \left(2\sum_{i=1}^n m_i\right)^{1/2} \right\}, \]
\[ \sum_{m=1}^{\infty} P\left\{ \max_{\bar x\in E_{e^m}}\xi(\bar x)> (2mn)^{1/2} + (2+\varepsilon)\ln(mn)/(2mn)^{1/2} \right\}, \]
\[ \sum_{m=1}^{\infty} P\left\{ \max_{\bar x\in E_{e^m}}\xi(\bar x)< \sqrt{2mn} - (2+\varepsilon)\ln(mn)/(2mn)^{1/2} \right\}. \]
Then the assertion of the theorem holds.
The convergence of the series can be verified by substituting the corresponding values for \(c\) in (1), (2), (3).
In conclusion, I express my sincere gratitude to M. I. Yadrenko for posing the problem and for his attention to this work.
Scientific Research Institute
of Construction Production
of Gosstroy of the Ukrainian SSR
Received
17 II 1970
REFERENCES
- H. Cramer, Bull. Am. Math. Soc., 68, 512 (1962).
- M. G. Shur, Theory of Probability and Its Applications, 10, 2 (1965).
- Yu. K. Belyaev, DAN, 176, No. 3, 495 (1967).