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UDC 513.88+519.21+519.53
MATHEMATICS
V. N. SUDAKOV
GAUSSIAN MEASURES, CAUCHY MEASURES, AND \(\varepsilon\)-ENTROPY
(Presented by Academician Yu. V. Linnik on 1 VII 1968)
Let \(x_t,\ t \in T\), be a Gaussian random process with zero mean and correlation function \(r(s,t)\). Let \(E\) be some Banach space of functions on \(T\). Below we give conditions which can often be checked from the function \(r(s,t)\) and which in most cases make it possible to decide whether the realizations of the process belong to the space \(E\).
- The correlation function \(r(s,t)\) determines an embedding \(T \subset \mathscr H\) of the set \(T\) into the Hilbert space \(\mathscr H\) of all measurable linear functionals over the Gaussian process \(\left({}^{1}\right)\). In what follows the separability of \(\mathscr H\) is always assumed. Consider the linear hull \(\mathscr L(T)\) and on it the norm induced by the duality \((E,\mathscr L(T))\); let \(K_1 \subset \mathscr L(T)\) be the unit ball. If \(E\) has full measure, then \(K_1\) is relatively compact in \(\mathscr H\) and the embedding \(K_1 \subset \mathscr H\) is continuous from \(\sigma(\mathscr L(T),E)\). Let \(K=\overline{K}_1\); \(\mathscr L(K)\subset \mathscr H\) is Banach and consists of all linear forms continuous on \(E\) and measurable with respect to the \(\sigma\)-algebra generated by the functionals from \(T\). In what follows the compact topology is considered on \(K\).
Proposition 1. \(E\) consists of linear forms bounded on \(K\), and contains the set of all linear forms continuous on \(K\).
Proposition 2. Let \(L\) be a linear space with Gaussian measure \(\mu\), and let \(L_1 \subset L\) be a measurable linear subspace. Then either \(\mu L_1=0\), or \(\mu L_1=1\) (see \(\left({}^{2}\right)\)).
The process \(x_t\) is stochastically continuous with respect to the metric on \(T\) induced from \(\mathscr H\), and therefore its realizations are measurable. The set \(K\) is restored from \(T\subset \mathscr H\). In particular, if \(E\) is a space of functions with the uniform norm, then \(K\) is the closed convex hull of \(T\). In any case, we arrive at the following general scheme. A Banach space \(E\) and another normed space \(E_1\) in duality with \(E\) are given. On \(E\) a separable Gaussian weak distribution (a generalized random process) is given, i.e., a mapping (which we may regard as an embedding) of \(E_1\) into the subspace \(\mathscr H\) of Gaussian-distributed functions of the space \(S_\mu\) of all measurable functions on some set with a separable measure \(\mu\). This embedding is bounded and continuous from the topology \(\sigma(E_1,E)\), and the unit sphere \(K_1 \subset E_1\) is compact in \(\mathscr H\). The problem is to find, in terms of \(K_1\subset \mathscr H\), conditions for the extendability of the weak distribution to a measure in \(E\).
- Conditions for the possibility of extending a weak Gaussian distribution to a measure, close to necessary and sufficient ones, can be given in terms of the \(\varepsilon\)-entropy of the compact set \(K\subset \mathscr H\). The entropy type (see, for example, \(\left({}^{3}\right)\)) of the compact set \(K\) is the number
\[ \rho(K)=\limsup_{\varepsilon\to 0} \frac{\log H(K;V_{\mathscr H},\varepsilon)}{\log 1/\varepsilon}, \]
where \(H(K;V_{\mathscr H},\varepsilon)=\log N(K;V_{\mathscr H},\varepsilon)\) is the \(\varepsilon\)-entropy of \(K\), and \(V_{\mathscr H}\) is the unit ball of \(\mathscr H\). Let \(K=\overline{K}_1\).
Theorem 1. If \(\rho(K)>2\), then the realizations of the process \(x_t,\ t\in K\), are unbounded with probability 1.
Theorem 2. If \(\rho(K)<2\), then the realizations of the process \(x_t,\ t\in K\), are bounded with probability 1.
Consider the weak Cauchy distribution whose characteristic functional on an element \(h \in \mathscr H\) is equal to \(\exp(-\|h\|_{\mathscr H})\). Since
\[ \exp(-r)=\int_0^\infty \exp\left(-\frac12(r\sigma)^2\right)\left(\sqrt{\frac{2}{\pi}}\frac{1}{\sigma^2}\exp\left(-\frac{1}{2\sigma^2}\right)\right)\,d\sigma, \tag{1} \]
and \(\exp(-\frac12\|h\|_{\mathscr H})\) is the characteristic functional of our Gaussian distribution, the stock of linearly measurable sets of full measure for the Cauchy measure \(\varkappa\) and the Gaussian measure \(\mu\) is one and the same. The conditional measures \({}^{(4)}\) of the measure under a (measurable) partition of the linear space with Gaussian measure \(\mu\) into rays are \(\delta\)-measures on almost every ray (cf. Shneiberg’s theorem \({}^{(5)}\)), and formula (1) shows that for the Cauchy measure \(\varkappa\) the conditional measures have density
\[ p(\sigma)=\sqrt{\frac{2}{\pi}}\,\frac{1}{\sigma^2}\exp\left(-\frac{1}{2\sigma^2}\right). \]
For any measurable \(A\),
\[ \varkappa A=\int_0^\infty \mu^\sigma A\,p(\sigma)\,d\sigma, \]
where \(\mu^\sigma\) has characteristic functional \(\exp\left(-\frac12(\|h\|\sigma)^2\right)\).
Proposition 3 (Shleffli–Slepian theorem \({}^{(6,7)}\)). Let \(A,B \subset \mathscr H\) and suppose there exists a mapping \(\psi\) of the set \(A\) onto \(B\) such that for \(f,g \in A\)
\[ \frac{(f,g)+1}{(\|f\|^2+1)^{1/2}(\|g\|^2+1)^{1/2}} \le \frac{(\psi f,\psi g)+1}{(\|\psi f\|^2+1)^{1/2}(\|\psi g\|^2+1)^{1/2}}. \]
Then \(\varkappa A^0 \le \varkappa B^0\), where \(C^0\) is the set of linear forms from the space with measure that do not exceed \(C\) in absolute value on \(C\).
Choose an orthonormal basis \(\{e_k\}\) in \(\mathscr H\). By a parallelepiped \(\pi(\{a_k\})\) with parameters \(a_k \downarrow 0\) we shall mean the set
\[ \pi=\{h:\sup a_k^{-1}|(h,e_k)|\le 1\}. \]
Let \(\hat\pi\) be the set of vertices of \(\pi\). A \(GB\)-set is \({}^{(8)}\) a set \(A \subset \mathscr H\) for which \(\mu(\lambda A^0)>0\) for some \(\lambda>0\).
Proposition 4. In the class of parallelepipeds the condition \(\rho(\hat\pi)\le 2\) is necessary for the \(GB\)-property. Indeed, it can be shown (cf. \({}^{(10)}\)) that \(H(\varepsilon)=H(\hat\pi(\{a_k\});V_{\mathscr H}\varepsilon)\) for small \(\varepsilon\) is equivalent to the number of terms of the sequence
\[ \left\{\left(\sum_{k=n}^{\infty} a_k^2\right)^{1/2}\right\} \]
greater than \(\varepsilon\). Therefore \({}^{(9)}\) the quantity
\[ \rho(\hat\pi)=\limsup \left(\frac{\log H(\varepsilon)}{\log 1/\varepsilon}\right) \]
is equal to the exponent of convergence of this sequence:
\[ \rho(\hat\pi)=\inf\left\{\alpha:\sum_n \left[\left(\sum_{k=n}^{\infty} a_k^2\right)^{1/2}\right]^\alpha<\infty\right\}. \]
If now \(\rho(\hat\pi)>2\), then \(\sum n a_n^2=\infty\), and hence \(\sum a_n=\infty\), i.e. \(\pi\) is not a \(GB\)-set (the three-series theorem).
Proof of Theorem 1. Let \(\{h_j^{(k)}\}, j=1,\ldots,M_k\), be a sequence of \(\varepsilon_k\)-nets \(K\), for which
\[ \lim \frac{\log\log M_k}{\log 1/\varepsilon}>2. \]
For some \(c=c(K)\), by Proposition 3 we obtain
\[ \varkappa K^0 \le \varkappa\{c\varepsilon_k e_1,\ldots,c\varepsilon_k e_{M_k}\}^0. \]
A direct calculation verifies that
\[ \mu^\sigma\{c\varepsilon_k e_1,\ldots,c\varepsilon_k e_{M_k}\}^0 \to 0 \]
for any \(\sigma>0\), and then also
\[ \varkappa\{c\varepsilon_k e_1,\ldots,c\varepsilon_k e_{M_k}\}^0 \to 0. \]
Since \(\varkappa K^0=0\), it follows that also \(\varkappa \mathscr L(K^0)=0\) and \(\mu \mathscr L(K^0)=0\).
Proof of Theorem 2. Let \(\rho(K)<2\). Consider a sequence \(\{h_j^{(k)}\}, j=1,\ldots,N(2^{-k})\), of minimal \(2^{-k}\)-nets of \(K\), and let \(N_{k_0}=N(2^{-k_0})=1\), while \(N_{k_0+1}>1\). Let
\[ \bar H_k=-[-\log N_k]. \]
Construct a parallelepiped \(\pi=\pi(\{a_k\})\), for which the first \(\bar H_{k_0+1}\) terms are equal to \(2^{-k}c\), the next \(\bar H_{k_0+2}\) terms are each equal to \(2^{-k-1}c\), etc. For this parallelepiped \(\rho(\hat\pi)=\rho(K)\), and there will be found such \(c=c(K)\) and such a subset of the set \(\pi\) that for it there exists a “contracting,” in the sense of Proposition 3, mapping onto the union of all the chosen \(\varepsilon\)-nets. Thus
\[ \varkappa K^0 \ge \varkappa \pi^0>0, \]
i.e., indeed,
\[ \mu\mathscr L(K^0)=1. \]
- Proposition 5. If \(Q\) is a set measurable with respect to the point \(O\) and
\[ q=\int_0^1 \sqrt{\frac{2}{\pi}}\,\frac{1}{\sigma^2}\exp\left(-\frac{1}{2\sigma^2}\right)d\sigma \approx 0.3174, \]
then
\[ \mu Q \ge (\varkappa Q-q)/(1-q). \]
The proof follows from the formula for conditional Cauchy measures.
Corollary (Dudley’s theorem \((^8)\)). If \(\rho(K)<2\), then the space \(E_0(K)\) of linear forms continuous on \(K\) has full measure.
Indeed,
\[ E_0(K)=\bigcap_{\lambda>0}\lambda\bigcup_n (K\cap L_n)^0, \]
where \(L_n\) is a decreasing sequence of closed subspaces with zero intersection. For any \(\lambda\), if \(n\) is sufficiently large, then \(\varkappa(\lambda(K\cap L_n)^0)\), as follows from the proof of Theorem 2, is sufficiently close to 1, and then, by Proposition 4, also \(\mu(\lambda(K\cap L_n))^0\).
Proposition 6. If the measure of the space \(E_0(K)\) of continuous forms is zero, then for some \(\varepsilon>0\) the realizations with probability 1 are not bounded by the number \(\varepsilon\).
Indeed, by the zero-one law, \(\mu\left(\lambda\bigcup_n (K\cap L_n)^0\right)\) is then equal to zero for some \(\lambda>0\).
Let us summarize the conclusion from Theorems 1 and 2, Proposition 1, and the corollary:
Theorem 3. Suppose that in a Banach space \(E\) a weak Gaussian distribution is given by means of a set of linear functionals \(E_1\), with a characteristic functional \(\chi(x)=\exp\left(-\frac12 A(x,x)\right)\) that is continuous in norm on \(E_1\). In order that this weak distribution extend to a measure, it is necessary that the unit ball \(K_1\subset E_1\), in the sense of duality \((E,E_1)\), have, in the norm \(\|x\|_{\mathscr H}^2=A(x,x)\), entropy type \(\rho(K_1)\le 2\), and it is sufficient that it have entropy type \(\rho(K_1)<2\).
With the aid of this theorem one can obtain, for example, the well-known Hunt–Belyaev conditions \((^{11})\) for continuity of realizations of a stationary Gaussian process.
- A necessary and sufficient condition for an ellipsoid \(K\) to possess the \(GB\) property is the condition
\[ \int_0^1 \varepsilon^2 dH(\varepsilon)>-\infty. \]
It seems implausible that, in terms of \(\varepsilon\)-entropy, there should exist an exhaustive answer in the general case. Let, for example, \(\pi=\pi(\{1/k\ln k\})\).
Then \(\pi\notin GB\), but
\[ \int_0^1 \varepsilon^2 dH(\pi; V_{\mathscr H},\varepsilon)>-\infty, \]
therefore, for an ellipsoid with the same growth \(H\) (there exists an ellipsoid \(K\) for which \(H_\pi(a\varepsilon)\le H_k(\varepsilon)\le H_\pi(b\varepsilon)\)) the \(GB\) property is satisfied. Passing to a sequence of finite-dimensional sections also does not improve the situation.
- The results of items 2 (Theorems 1 and 2) and 4 were reported by the author at the International Congress of Mathematicians in 1966. Recently Dudley’s paper \((^8)\) appeared, in which he notes that Theorem 2 had also been proved by Strassen (unpublished), and he himself finds a certain strengthening of this theorem, which can also be obtained by our methods (a corollary of Proposition 5). The notation for the \(GB\) property belongs to him as well \((^8)\).
The author expresses his gratitude to A. M. Vershik, I. A. Ibragimov, and B. M. Makarov for useful discussions.
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
17 VI 1968
REFERENCES
- A. M. Vershik, DAN, 170, No. 3 (1966).
- V. N. Sudakov, Some problems connected with distributions in infinite-dimensional linear spaces, Dissertation, LSU, 1962.
- A. Dvoretzky, B. Mitiagin, Bull. Acad. Polon. Sci., ser. math., astr. et phys., 8, No. 7 (1960).
- V. A. Rokhlin, Mat. sborn., 25 (67), No. 1 (1949).
- J. Schoenberg, Ann. Math., 39 (1938).
- L. Schläfli, Quart. J. Pure and Appl. Math., 2, 1858, 3 (1860).
- D. Slepian, Bell System Techn. J., 16, No. 2 (1962).
- R. M. Dudley, J. Functional Analysis, 1, No. 3 (1967).
- B. Ya. Levin, Distribution of zeros of entire functions, 1956.
- B. S. Mityagin, UMN, 16, issue 4 (100) (1961).
- Yu. K. Belyaev, Theory of Probability and Its Applications, 5, issue 1 (1960).
- A. Kolmogorff, Ann. Math., 37, 107 (1936).