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UDC 519.21
MATHEMATICS
E. I. OSTROVSKII
ON THE LOCAL STRUCTURE OF NORMAL FIELDS
(Presented by Academician A. N. Kolmogorov, 15 IV 1970)
Let \(T\) be a separable compact topological space with the second axiom of countability; \((\Omega, B, P)\) the basic probability space; \(x_t\), \(t \in T\), a separable normal field whose correlation function \(R(t,s)\) is continuous: \(R: T \otimes T \to R^1\). It is also assumed that on \(T\) there exists a finite measure \(\mu\), regular with respect to the topology \(T\); we shall denote the topology by \(\tau\): \(\mu(T)=1\). \(C\) is the space of all continuous functions mapping \(T\) into \(R^1\), with norm \(\forall f \in C\)
\[
\|f\|=\sup_{t\in T}|f(t)|;
\]
\(L_2\) is the space of functions quadratically summable with respect to \(\mu\), with norm
\[
\|f\|_{L_2}^2=\int_T f^2(t)\,d\mu(t).
\]
Denote by \(\lambda_k\) and \(\varphi_k(t)\), respectively, the eigenvalues and eigenfunctions of the operator \(R: L_2 \to L_2\), given by the formula
\[
(Rf)(t)=\int_T f(s)R(t,s)\,d\mu(s),
\]
where \(\|\varphi_n\|_{L_2}=1\); by the theorems of Mercer and Hilbert–Schmidt,
\[
\varphi_k \in C;\qquad \lambda_k \geq 0;\qquad (k=1,2,\ldots),
\]
\[
R(t,s)=\sum_{k=1}^{\infty}\lambda_k\varphi_k(t)\varphi_k(s),
\tag{1}
\]
where the series in (1) converges absolutely and uniformly (see \((^2)\), p. 250). Put
\[
\xi_k=\frac{1}{\sqrt{\lambda_k}}\int_T x_t\varphi_k(t)\,d\mu(t),
\tag{2}
\]
\(\xi_k\) are independent, normal with parameters \((0,1)\). Denote
\[
S_n(t)=\sum_{k=1}^{n}\sqrt{\lambda_k}\xi_k\varphi_k(t).
\]
Theorem 1.
\[
P\left\{\forall t\in T,\ \lim_{n\to\infty}S_n(t)=x_t\right\}=1.
\tag{3}
\]
Theorem 2. The set of discontinuity points of \(x_t\) is one and the same for almost all trajectories \((^3)\).
Theorem 3. In any neighborhood of a discontinuity point, with probability 1, the trajectory is unbounded.
The proof essentially repeats the analogous proof of Yu. K. Belyaev (\((^1)\), pp. 23–33), if instead of the spectral expansion one uses Theorem 1 and the following lemma.
Lemma 1. Let \(t_0\) be a discontinuity point of \(x_t\). Then there exists a nonrandom constant \(a>0\) such that
\[ P\left\{\overline{\lim_{t\to t_0}} |x_t-x_{t_0}|>2a\right\}=1. \]
Proof. By Theorem 2 the process \(x_t\) is discontinuous at \(t_0\) with probability 1; this means that
\[ P\left\{\overline{\lim_{t\to t_0}} |x_t-x_{t_0}|=\varepsilon\right\}=1,\qquad \varepsilon>0, \]
but then the quantity \(\varepsilon=\varepsilon(\omega)\) is measurable with respect to the smallest \(\sigma\)-algebra generated by the quantities \(\xi_k,\xi_{k+n},\ldots,\ \forall n\ge 1\), since \(S_\xi(t)\) is continuous with probability 1; hence there exists a constant \(a>0\) such that \(P\{\varepsilon=2a\}=1\), as was required to prove.
Corollary. Let \(G\) be a subgroup of the group of homeomorphisms of \(T\) onto itself such that for all \(g\in G\)
\[ R(tg,t_1g)=R(t,t_1). \]
Then the set of discontinuity points of \(x_t\) is invariant with respect to \(G\). If, moreover, \(T\) is full with respect to \(G\), then the process \(x_t\), if it is discontinuous, is unbounded in every open set.
To derive a necessary and sufficient condition for continuity, suppose that \(T=[0,2\pi]\); without essential loss of generality, \(x_0=x_{2\pi}\). Let \(f(x)\in C\) and be periodic; put, for \(p\ge 1\), \(p\) an integer,
\[ \omega_p(f,\delta)=\sup_{|h|<\delta}\sup_x \left|\sum_{k=0}^{p}(-1)^{p-k} C_p^k f(x+kh)\right|, \]
\[ K_n(x)=\frac{3}{2\pi n(2n^2+1)}\sin^4\left(\frac{nx}{2}\right)\bigg/\sin^4\left(\frac{x}{2}\right) \]
(the Jackson kernel \((^4)\), pp. 140–145)
\[ K_{mn}(x)=K_m(x)-K_n(x); \]
\[ L_{mn}^k(t)=\int_{-\pi}^{\pi} e^{isk}K_{mn}(t-s)\,ds; \]
\(C^{2m+1}\) will denote the \((2m+1)\)-dimensional complex-number space with coordinates \(\forall z\in C^{2m+1}\)
\[ z=(z_{-m};z_{-m+1};\ldots z_{-1};z_0;z_1;\ldots z_m), \]
and define
\[ (z,z)=\sum_{q=-m}^{m}|z_q|^2;\qquad dz=\prod_{q=-m}^{m} d\operatorname{Re} z_q\, d\operatorname{Im} z_q. \]
The Bernstein function \(\Phi(z)\) is defined for \(z\in C^{2m+1}\) by the equality (see \((^5)\), p. 127 ff.)
\[ \Phi(z)=\sup_x \left|\sum_{k=-m}^{m} z_k e^{ikx}\right|. \]
$\tilde R_m^n$ is a square matrix of order $(2m+1;\,2m+1)$ with elements
\[ (\tilde R_m^n)_{kl}=\int_0^{2\pi}\int_0^{2\pi} L_{mn}^k(t)\,\overline{L_{mn}^l(s)}\,R(t,s)\,dt\,ds \]
and $R_m^n$ is the Hermitian square root of $\tilde R_m^n$. Also put, for $m>n$,
\[ \tau_m^n=(2\pi)^{-2m}\int_{C^{2m+1}}\operatorname{arctg}\Phi(R_m^n z)e^{-1/2(z,z)}\,dz. \]
From known theorems of constructive function theory one can derive the following result ([4], pp. 144–145).
Theorem 4. In order that $x_t$ be continuous with probability 1, it is necessary and sufficient that
\[ \lim_{n\to\infty}\sup_{m>n}\tau_m^n=0. \]
Theorem 5. Suppose that the condition of Theorem 4 is satisfied; then there exist constants $C_p$, $d_p$ such that $\forall p\geqslant 1$
\[ C_p\sup_{m>n}\tau_m^n\leqslant M\operatorname{arctg}\omega_p\left(x_t,\frac{1}{n}\right) \leqslant \frac{d_p}{n^p}\sum_{\nu=1}^n \nu^{p-1}\sup_{m>\nu}\tau_m^\nu . \]
By the indicated method one easily obtains the known sufficient conditions for continuity of the process $x_t$, if one uses the known Bernstein inequality ([4], p. 146).
Put
\[ \tilde\omega_R(\delta)=\sup_{\substack{|h_1|\leqslant\delta\\ |h_2|\leqslant\delta}} \sqrt{\left|R(t+h_1,s+h_2)-R(t,s)\right|}. \]
Theorem 6 (Fernique, [6]). If
\[ \int_0^\infty \tilde\omega_R(e^{-x^2})\,dx<\infty, \]
then $x_t$ is continuous with probability 1.
Corollary. Let $\omega_1(\delta)$ be a modulus of continuity such that
\[ \varlimsup_{\delta\downarrow 0}\frac{\omega_1(\delta)}{\delta}=\infty . \]
Then there exists a stationary process $x_t$ such that for its correlation function $r(t)=Mx_s x_{s+t}$, $\omega_1[r(t),\delta]=\omega_1(\delta)$, but at the same time $P\{x_t\in G\}=1$; and even $M\exp[\varepsilon\|x_t\|^2]<\infty$ for some $\varepsilon>0$.
Theorems 4 and 5 have also been obtained for finite-dimensional fields.
The author takes this opportunity to express gratitude to S. A. Molchanov for supervising the work and to Yu. K. Belyaev for assistance and valuable suggestions.
Moscow State University
named after M. V. Lomonosov
Received
7 III 1970
CITED LITERATURE
- Yu. K. Belayev, Proc. IV Berkeley Symp. on Math. Statistics and Probability, California, 1960, 2, Berkeley, 1961, p. 23.
- N. Dunford, J. Schwartz, Linear Operators, 2, IL, 1962.
- T. Kawada, Proc. Soviet-Japanese Symposium on Probability Theory, Khabarovsk, 1969, p. 105.
- R. S. Guter, L. D. Kudryavtsev, B. M. Levitan, Elements of Function Theory, Moscow, 1963.
- S. N. Bernstein, Collected Works, 2, Publishing House of the Academy of Sciences of the USSR, 1954.
- X. Fernique, C. R., 258, 6058 (1964).