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1 Introduction↩︎

Let \((\Omega,\mathcal{A},\mu)\) be a complete probability space and \(\xi : T \times \Omega \to \mathbb{R}\) a centered Gaussian process with covariance function \(K\). For such a random process, questions concerning the regularity of its sample paths arise immediately. A classical result asserts that the sample paths of \(\xi\) belong to the reproducing kernel Hilbert space \(H(K)\) of \(K\) with probability zero whenever \(H(K)\) is infinite-dimensional. This result was originally stated without proof by Parzen [1], and later proved by Kallianpur [2] and Le Page [3].

This motivated Driscoll [4] to ask whether there exists another reproducing kernel Hilbert space \(H(R)\) that contains the sample paths of \(\xi\) almost surely. Under appropriate assumptions, Driscoll proved the zero–one law \[\label{eq:Driscoll:zero:one} \mu(\xi \in H(R)) = 0 \qquad \mathrm{or} \qquad \mu(\xi \in H(R)) = 1,\tag{1}\] and established a necessary and sufficient condition for \(\mu(\xi \in H(R)) = 1\) to hold. The condition given by Driscoll is equivalent to requiring that \(H(K) \subseteq H(R)\) and that the operator \(L : H(R) \to H(R)\), defined by \(L R(\cdot,t) = K(\cdot,t)\) for all \(t \in T\), is nuclear. Fortet [5] subsequently introduced the terminology that \(R\) dominates \(K\) if \(H(K) \subseteq H(R)\), and that \(R\) n-dominates \(K\) if, in addition, \(L\) is nuclear. In modern terminology, this is commonly referred to as nuclear dominance.

The assumptions made by Driscoll in the original paper are that \(T\) is a separable metric space and that the sample paths of \(\xi\) are continuous with respect to the metric topology on \(T\). Driscoll describes the latter condition as “a rather restrictive assumption, even for Gaussian processes” [4]. Nevertheless, this assumption allows \(\xi\) to be regarded as a random element taking values in the space \(C(T)\) of continuous functions on \(T\), thereby addressing a fundamental measurability issue that arises for random processes and the cylindrical \(\sigma\)-algebra on \(\mathbb{R}^T\).

Lukić and Beder [6] later proved that none of Driscoll’s additional assumptions are necessary. The crux of their analysis is the canonical metric on \(T\) induced by the kernel \(R\), with respect to which functions in \(H(R)\) are continuous. Under the nuclear dominance assumption, \(H(R)\) may be assumed to be separable, and hence \(T\) admits a countable dense subset. Random elements in \(H(R)\) can therefore be defined by specifying them on such a subset and extending them uniquely by continuity. This yields a natural candidate for a suitable modification of the original random process \(\xi\) taking values in \(H(R)\). Thereby the measurability difficulties that motivated Driscoll’s additional assumptions can be resolved by reducing the problem to this countable dense subset of \(T\), where these difficulties do not arise.

In this paper, we extend the analysis of Lukić and Beder [6] to the Banach space setting and generalize Driscoll’s theorem to separable reproducing kernel Banach spaces. Our main results are Theorems 23 and 24, which establish the “one” and “zero” parts of Driscoll’s theorem, respectively. These theorems are summarized in the following theorem using \(\gamma\)-radonifying operators [7], [8]. The \(\gamma\)-radonifying operators can be thought of as operators that map “white noise” in a Hilbert space into a well-defined Gaussian random element in a Banach space (see Theorem 13).

Similar to Lukić and Beder [6], a cornerstone of our analysis is the connection between random elements in reproducing kernel Banach spaces and the associated stochastic processes. We show that the covariance operator of a (weak) second-order Radon probability measure on such a space is uniquely determined by a positive definite function. In particular, we characterize those positive definite functions that arise as covariance functions of Radon Gaussian measures. This result replaces the Prokhorov theorem used in Driscoll’s original paper [4]. Moreover, our analysis provides a perspective on kernel-based methods that is more natural when considering pushforwards under measurable mappings between Banach spaces.

Several authors have established \(\gamma\)-radonifying embeddings of reproducing kernel Hilbert spaces associated with Gaussian processes into classical function spaces. A fundamental result due to Brzeźniak [9] shows that the indefinite integration operator is \(\gamma\)-radonifying as an operator from \(L^2(0,1)\) into the fractional Sobolev space \(W^{\alpha,p}(0,1)\) for \(p \in (2,\infty)\) and \(\alpha \in \left(1/p, 1/2\right)\). This yields fractional Sobolev regularity, and hence Hölder regularity, of Brownian motion. The results in [9] are formulated in terms of almost sure convergence of Gaussian series rather than in the language of \(\gamma\)-radonifying operators. Further details, refinements, and results on Besov regularity can be found in [9] and the references therein.

2 Preliminaries↩︎

Here we recall some basic definitions and properties of random elements in locally convex spaces and their distributions. Further details and proofs can be found in the classical texts [10], [11]. Throughout this section, \(X\) denotes a locally convex space, \(X^*\) its topological dual, and \((\Omega,\mathcal{A})\) a measurable space. We write \(\langle \cdot, \cdot \rangle\) for the dual pairing between \(X\) and \(X^*\) defined by \(\langle x ,f \rangle = f(x)\) for \(x \in X\) and \(f \in X^*\).

The \(\sigma\)-algebra generated by a family \(\Gamma \subseteq X^*\) is called the cylindrical \(\sigma\)-algebra with respect to \(\Gamma\) and will be denoted by \(\mathcal{C}(X,\Gamma)\). It is the smallest \(\sigma\)-algebra on \(X\) with respect to which every \(f \in \Gamma\) is measurable. In the special case \(\Gamma = X^*\), we write \(\mathcal{C}(X)\). The Borel \(\sigma\)-algebra on \(X\) will be denoted by \(\mathcal{B}(X)\). In some important cases \(\mathcal{B}(X)\) is strictly larger than \(\mathcal{C}(X,\Gamma)\). However, if \(X\) is a separable Fréchet space and \(\Gamma\) separates points the \(\sigma\)-algebras coincide.

An essential analytical tool in the study of probability measures on locally convex spaces is the characteristic functional, also called the Fourier transform.

By the Lebesgue dominated convergence theorem, \(\widehat{\mu}\) is sequentially continuous on \(\Gamma\) endowed with the weak-\(*\) topology. In general, however, continuity of \(\widehat{\mu}\) requires a stronger topology on \(\Gamma\), such as the topology of uniform convergence on compact subsets of \(X\), or the Mackey topology if \(X\) is quasi-complete.

An important property of the characteristic functional is that if two probability measures \(\mu_1\) and \(\mu_2\) on \(\mathcal{C}(X,\Gamma)\) satisfy \(\widehat{\mu}_1 = \widehat{\mu}_2\), then \(\mu_1 = \mu_2\).

It is well known that a probability measure \(\gamma\) on \((X,\mathcal{C}(X))\) is Gaussian if and only if, for every \(f \in X^*\), the pushforward \(\gamma \circ f^{-1}\) is a Gaussian measure on \((\mathbb{R}, \mathcal{B}(\mathbb{R}))\). Indeed, many authors take this property as the definition of a Gaussian measure.

For \(1 \leq p < \infty\), let \(L^p(\mu)\) denote the Lebesgue space of \(p\)-integrable functions with respect to the measure \(\mu\). A probability measure \(\mu\) is said to be of weak \(p\)-order if \(X^* \subseteq L^p(\mu)\). For probability measures of weak order \(p \geq 2\), the mean and covariance can be defined as follows.

It follows from Fernique’s theorem that every Gaussian measure is of weak order \(p\) for all \(p \in [1,\infty)\). For a Radon Gaussian measure, the mean can be represented by an element of \(X\) and the covariance by a continuous linear operator \(R : X^* \to X\). We recall that a measure \(\mu\) on \((X, \mathcal{B}(X))\) is called a Radon measure if, for every \(B \in \mathcal{B}(X)\) and \(\varepsilon > 0\), there exists a compact set \(K \subseteq B\) such that \(\mu(B \setminus K) < \varepsilon\).

The concept of a Gaussian measure is closely related to that of Gaussian random processes. Let \(T\) be a non-empty set, and let \(\mathbb{R}^T\) be the space of real-valued functions on \(T\), endowed with the topology of pointwise convergence.

A straightforward calculation shows that \(\xi : T \times \Omega \to \mathbb{R}\) is a random process if and only if the map \(\omega \mapsto \xi(\cdot,\omega)\) is \((\mathcal{A}, \mathcal{C}(\mathbb{R}^T))\)-measurable. Hence a random process can equivalently be regarded as a random element in \((\mathbb{R}^T, \mathcal{C}(\mathbb{R}^T))\). For \(\omega \in \Omega\), the realization \(\xi(\cdot,\omega)\) is called a sample path, and \(\mathbb{R}^T\) is called the path space.

For \(t \in T\), let \(\delta_t:\mathbb{R}^T\to\mathbb{R}, \, x \mapsto x(t)\) be the evaluation functional at \(t\). Using the characterization \((\mathbb{R}^T)^* = \operatorname{span}\{ \delta_t : t \in T \}\), it follows that a random process \(\xi: T \times \Omega \to \mathbb{R}\) is Gaussian if and only if \((\xi_t)_{t \in T_0}\) is a Gaussian random element for every finite subset \(T_0 \subseteq T\).

Because \((\mathbb{R}^T)^* = \operatorname{span}\{ \delta_t : t \in T \}\) and \(R_\gamma\) is bilinear, it follows that \(R_\gamma\) is uniquely determined by the function \(K\) in Proposition 6. Accordingly, \(K\) is called the covariance function of \(\gamma\). It is readily verified that \(K\) is positive definite.

It is well known that a positive definite function \(K : T \times T \to \mathbb{R}\) is automatically symmetric, in the sense that \(K(s,t) = K(t,s)\) for all \(s,t \in T\). Accordingly, when referring to a positive definite function, we omit the qualifier “symmetric.”

A remarkable property of Gaussian processes is that every positive definite function determines a unique (centered) Gaussian process. This follows immediately from the celebrated Kolmogorov extension theorem. For Gaussian measures on a general locally convex space \(X\), however, describing the bilinear forms that arise as Gaussian covariances is much more delicate.

3 Covariance Operators↩︎

Throughout this section, \(X\) denotes a Banach space with norm \(\|\cdot\|\) and \(H\) denotes a Hilbert space with inner product \((\cdot,\cdot)\). Their dual spaces \(X^*\) and \(H^*\) are endowed with the respective norm topologies unless otherwise specified. We write \(\mathcal{L}(H,X)\) for the space of bounded linear operators from \(H\) to \(X\), endowed with the strong operator topology. When there is a risk of confusion, we add a subscript to specify the underlying space; for example, \(\|\cdot\|_{X}\) denotes the norm on \(X\).

It is readily verified that every covariance operator is symmetric and positive. Therefore we begin by recalling some of their basic properties. Proofs of the results cited can be found in [11].

Because every symmetric operator is automatically continuous and linear, we will omit these qualifiers when referring to symmetric operators.

From the Cauchy–Schwarz inequality it follows that every symmetric positive operator \(R : X^* \to X\) naturally induces an inner product on its range \(R(X^*)\) with \[\label{eq:cameron:martin:inner} (Rf, Rg) = \langle Rf, g \rangle, \qquad f,g \in X^*.\tag{8}\] In general, \(R(X^*)\) is not complete with respect to the norm induced by the inner product. However, using that \(X\) is (sequentially) complete, one obtains a Hilbert space completion of \(R(X^*)\) as a subspace of \(X\).

The Hilbert space \(H\) in Proposition 8 is called the Hilbert space associated with \(R\) and will be denoted by \(H(R)\). Because \(R(X^*)\) is a dense subspace of \(H(R)\) and the embedding \(\iota : H(R) \to X\) is continuous, it follows from 8 that the inner product on \(H(R)\) satisfies the reproducing property \[\label{eq:reproducing:property} (x, Rf) = f(x), \qquad x \in H(R),\;f \in X^*.\tag{9}\]

The following factorization lemma shows that \(R\) is uniquely determined by \(H(R)\). For an operator \(A \in \mathcal{L}(H,X)\) we denote by \(A^{*}\) its adjoint, and we identify \(H\) with its dual \(H^{*}\) via the Riesz representation theorem.

It is clear that the converse of Lemma 2 also holds: if \(H\) is a Hilbert space and \(A \in \mathcal{L}(H,X)\), then \(R = A A^{*}\) is a symmetric positive operator. In this case, \(H\) is isometrically isomorphic to \(H(R)\), and the factorization is unique in this sense.

For a Radon Gaussian measure \(\gamma\) on \((X,\mathcal{B}(X))\) with covariance operator \(R\), it is well known that \(H(R)\) is separable. Hence \(R\) admits the representation 10 . This representation naturally suggests decomposing \(\gamma\) as a series expansion in the orthonormal basis \((x_k)\), with independent standard Gaussian coefficients.

Throughout this section \((\Omega, \mathcal{A}, \mu)\) is a fixed probability space supporting a countable sequence \((\xi_k)\) of independent standard Gaussian random variables. Such a sequence is called a Gaussian sequence.

A series of the form 11 is refered to as a Gaussian series and is treated in detail in [7], [10], [11]. Gaussian series are important for characterizing covariance operators of Radon Gaussian measures. We present one such characterization in terms of the associated Hilbert space, formulated using \(\gamma\)-radonifying operators. A detailed treatment of \(\gamma\)-radonifying operators can be found in [7], [8].

For the next definition, we introduce the notation \(\mathcal{F}(H)\) for the collection of all finite orthonormal systems in \(H\).

The collection of \(\gamma\)-summing operators will be denoted by \(\gamma_\infty(H,X)\). It is readily verified that every \(A \in \gamma_\infty(H,X)\) is bounded by considering orthonormal systems consisting of a single element. Moreover, \(\gamma_\infty(H,X)\) can be equipped with a norm for which it becomes a Banach space that is continuously embedded into \(\mathcal{L}(H,X)\).

For finite-rank operators the \(\gamma_\infty\)-norm admits an explicit expression. Recall that an \(A \in \mathcal{L}(H,X)\) has finite rank if its range \(A(H)\) is finite-dimensional, or equivalently, if there exists an orthonormal system \((h_k)_{k=1}^n \subset H\) such that \[\label{eq:finite:rank} A h = \sum_{k=1}^n (h, h_k)\, A h_k, \qquad \forall h \in H.\tag{14}\]

The closure of the subspace of finite rank operators in \(\gamma_\infty(H,X)\) will be denoted by \(\gamma(H,X)\), and operators in \(\gamma(H,X)\) are called \(\gamma\)-radonifying. The two spaces coincide when \(X\) does not contain a closed subspace isomorphic to \(c_0\) (the Banach space of null sequences in \(\mathbb{R}\)). In general, however, \(\gamma(H,X)\) is a proper subspace.

It is immediate that \(\gamma\)-radonifying operators can be approximated in \(\gamma(H,X)\) by finite rank operators. Since \(\gamma(H,X)\) is continuously embedded into \(\mathcal{L}(H,X)\), it follows that every \(\gamma\)-radonifying operator is compact and supported on a separable subspace. Moreover, \(\gamma\)-radonifying operators admit the following characterization.

It is straightforward to verify that, in the special case where \(X\) is a Hilbert space and \(A \in \mathcal{L}(H,X)\), the Gaussian series 16 converges almost surely if and only if \(A\) is a Hilbert–Schmidt operator. Recall that \(A\) is Hilbert–Schmidt if \[\label{eq:hilbert:schmidt} \sup_{(h_k)_{k=1}^n \in \mathcal{F}(H)} \sum_{k=1}^n \| Ah_k \|^2 < \infty.\tag{17}\]

Theorems 10 and 13 together characterize the covariance operators of Radon Gaussian measures. See also [8] and [11].

4 Reproducing Kernel Banach Spaces↩︎

Natural candidates for sample path spaces of random processes are function spaces that are continuously embedded into \(\mathbb{R}^T\). Such spaces are characterized by the continuity of pointwise evaluation.

If \(X\) is a Hilbert space it is called a reproducing kernel Hilbert space. Throughout this section, \(X \subseteq \mathbb{R}^T\) denotes a reproducing kernel Banach space with norm \(\|\cdot\|\), and \(\delta_t : X \to \mathbb{R}\) denotes the evaluation functional at \(t \in T\).

We show that covariance operators are uniquely determined by positive definite functions and characterize those positive definite functions that arise as kernels of Gaussian covariance operators. For a discussion of the role of reproducing kernel Banach spaces in machine learning, we refer to [14] and the references therein.

It is clear that the kernel of \(R\) is a well-defined positive definite function and that \(R \delta_t = K(\cdot,t)\) for all \(t \in T\). Moreover, since \(\Gamma = \{ \delta_t : t \in T \}\) separates points and \(R\) is symmetric, it follows that \(R\) is uniquely determined by its kernel \(K\). In analogy with random processes, when \(R\) is a covariance operator of a weakly second order probability measure \(\mu\) on \((X,\mathcal{C}(X))\) we call \(K\) the covariance function of \(\mu\).

Not every positive definite function determines a covariance operator of a Radon Gaussian measures, nor does it necessarily determine a symmetric positive operator. To obtain necessary and sufficient conditions for the existence of such an operator, we recall the celebrated Moore–Aronszajn theorem [15].

It follows from property ([thm:Moore:Aronszajn:reproducing]) that \(H\) is a reproducing kernel Hilbert space. For a positive definite function \(K:T \times T \to \mathbb{R}\), we denote the corresponding Hilbert space by \(H(K)\), and \(K\) is called the reproducing kernel of \(H(K)\). The notation and terminology are justified by the following results.

The previous lemma suggests that a symmetric positive operator factors through the Hilbert space associated with its kernel in the sense of Lemma 2. This leads to the following characterization of kernels of symmetric positive operators.

The previous proposition is the Banach space analogue of [6]. The condition \(H(K) \subseteq X\) is the dominance assumption, and the symmetric positive operator \(R\) induced by \(\iota\) is the corresponding dominance operator.

Although covariance operators of Radon Gaussian measures on \(X\) are nuclear operators, the converse is not true in the Banach space setting. Moreover, nuclear dominance concerns the operator \(R\) itself rather than the embedding \(\iota : H(K) \to X\). From the perspective of constructing (Radon) Gaussian measures on \(X\) from \(K\), it is arguably more natural to impose conditions on the embedding \(\iota\) rather than on the induced operator \(R\).

In the remainder of this section we establish several properties of reproducing kernel Banach spaces that will be used to address measurability issues for Gaussian processes by restricting to countable index sets. For a subset \(S \subseteq T\) and \(x \in \mathbb{R}^T\), we write \(x|_S\) for the restriction of \(x\) to \(S\).

For a subset \(S \subseteq T\), we denote by \(X(S)\) the restriction of \(X\) to \(S\) equipped with with the norm 19 , and by \(\pi_S : X \to X(S), \, x \mapsto x|_S\) the corresponding restriction map. A subset \(S\) is called a determining set for \(X\) if \(\pi_S\) is an isometric isomorphism. This terminology goes back to [5]. Theorem 19 yields the following equivalent conditions for \(S\) to be a determining set for \(X\).

To resolve measurability issues it is of principal interest to determine when a space \(X\) admits a countable determining set. A sufficient condition is provided by the following lemma. For the proof, recall that a subset \(\Delta \subseteq X^*\) is said to be weak-\(*\) sequentially dense in \(\Gamma \subseteq X^*\) if for every \(f \in \Gamma\) there exists a sequence \((f_k) \subseteq \Delta\) such that \(f_k \to f\) in the weak-\(*\) topology. Further details on weak-\(*\) sequential density and closures can be found in [17], [18].

When \(X\) is a Hilbert space with reproducing kernel \(K\), the converse also holds: if \(X\) has a countable determining set, then \(X\) is separable. In preparation for this result, we introduce the kernel metric on \(T\), defined by \[\label{eq:kernel:metric} d_K : T \times T \to \mathbb{R}, \qquad (s,t) \mapsto \| K(\cdot,s) - K(\cdot,t) \|.\tag{20}\] It is readily verified that \(d_K\) defines a pseudo-metric on \(T\). Moreover, by the reproducing property and the polarization identity, it follows that \[\label{eq:kernel:metric:Loeve} d_K(s,t) = \sqrt{K(s,s) + K(t,t) - 2K(s,t)}, \qquad \forall s,t \in T.\tag{21}\]

The following lemma extends [6] by including the equivalent condition ([lem:rkhs:separability:determining]). In [6], the proof of the implication ([lem:rkhs:separability:separable])\(\implies\)([lem:rkhs:separability:TdK]) relies on [19], which is stated without proof. We provide here a self-contained argument for this implication.

For reproducing kernel Banach spaces, a similar characterization is not available. Indeed, the following example shows that \((\ref{lem:rkhs:separability:determining}) \implies (\ref{lem:rkhs:separability:separable})\) may fail.

5 Driscoll’s Theorem↩︎

We are now in a position to establish the connection between Gaussian random elements in a reproducing kernel Banach space \(X\) and Gaussian random processes. In particular, we extend the results of Lukić and Beder [6] to the Banach space setting and provide necessary and sufficient conditions under which a Gaussian process defines a Gaussian random element in \(X\). While it is immediate that every Gaussian random element in \(X\) induces a Gaussian process, the converse is considerably more delicate and involves subtle measurability issues. The main results are Theorems 23 and 24, which together yield a Banach space generalization of the classical Driscoll theorem.

Throughout this section, \(X \subseteq \mathbb{R}^T\) denotes a separable reproducing kernel Banach space and \((\Omega,\mathcal{A},\mu)\) a probability space. For \(t \in T\), we write \(\delta_t : \mathbb{R}^T \to \mathbb{R},\, x \mapsto x(t)\), for the evaluation functional at \(t\). Because \(X\) is continuously embedded into \(\mathbb{R}^T\), we use the same notation for the restriction of \(\delta_t\) to \(X\). A random element \(\xi : \Omega \to X\) will, without further comment, also be regarded as a random process over \(T\). When \(\xi\) is viewed as a random element \(\mathbb{R}^T\) we write \(\xi_t = \langle \xi, \delta_t \rangle\) for its coordinate at \(t\).

We begin by addressing the subtle measurability issues arising when passing from a random process to an \(X\)-valued random element.

In particular, the lemma applies to \(C = X\), and hence \(X \notin \mathcal{C}(\mathbb{R}^T)\) whenever \(T\) is uncountable. As a consequence, there is no guarantee that \[\{ \omega \in \Omega : \xi(\cdot,\omega) \in X \} \in \mathcal{A}.\] To address the measurability issues arising from this fact, we avoid statements of the form “\(\xi\) has sample paths almost surely in \(X\)” and instead work with a measurable set \(\Omega_0 \in \mathcal{A}\) with \(\mu(\Omega_0) = 1\) such that \(\xi(\cdot,\omega) \in X\) for all \(\omega \in \Omega_0\). This is equivalent to \(X\) having full inner measure in the distribution \(\mu^\xi\) of \(\xi\).

The random element \(\eta\) is called the random element in \(X\) defined by \(\xi\). It is clear that \(\eta\) is uniquely defined up to a set of probability zero. A natural question is whether \(\eta\) is Gaussian whenever \(\xi\) is a centered Gaussian random process.

In [6], this is proved when \(X\) is a Hilbert space, using the weak-\(*\) sequential density of \(\Gamma = \operatorname{span}\{ \delta_t : t \in T \} \subseteq X^*\) in \(X^*\). However, when \(X\) is a Banach space, \(\Gamma\) need not be weak-\(*\) sequentially dense in \(X^*\) unless \(X\) is reflexive or, more generally, quasi-reflexive; see [18]. Therefore, to obtain a Banach space analogue, we instead appeal to the characterization of centered Gaussian measures through their rotational invariance.

Depending on the probability space \((\Omega,\mathcal{A},\mu)\), the assumption that \(X\) has full inner measure in Lemma 9 and Theorem 21 can be rather restrictive, as shown by Lemma 8. Although a random process may fail to satisfy this assumption, it may nevertheless admit a modification that does. This was proved in the Hilbert space setting in [6]. In preparation for extending this result to Banach spaces, we establish the following lemma.

For the proof of the Banach space analogue of [6], we follow the original approach of [4], replacing the Prokhorov theorem [11], which characterizes covariance operators of Gaussian measures on Hilbert spaces, with the corresponding Banach space version in Theorem 14.

The implication \((\ref{thm:Driscoll:one:radonifying}) \implies (\ref{thm:Driscoll:one:modification})\) in Theorem 23 may be viewed as the Banach space analogue of the “one” part of the Driscoll theorem. A similar statement appears in [20] in the setting of Hilbert spaces. It is formulated in a language predating reproducing kernel Hilbert spaces and nuclear dominance, which may explain why it is not cited in [6]. The corresponding “zero” part is proved similarly to [6], using the Kallianpur zero–one law [21].

The identity operator is \(\gamma\)-radonifying, and hence compact, precisely in finite-dimensional Hilbert spaces. As an immediate consequence, we obtain the following classical result, originally stated by Parzen [1] and subsequently proved by Kallianpur [2] and Le Page [3].

References↩︎