The first main theorem 15 6. Iterated function systems 28 9. Gaussian versus non-Gaussian 31 Boundaries of positive definite functions 39 References 45 1. Introduction We are considering three functional analytic questions arising at the crossroads of pure and applied probability theory.
This is easy enough in simple discrete models, for example in experiment with tossing coins. However in infinite models, and in most continuous models, a complete description of a sample space of outcomes and its subsets, events, presents subtle problems.
The F -measurable functions are random variables, and systems of random variables are stochastic processes. As a result, there is a need for discretizations. Several such discretizations will be presented here, and comparisons will be made. Our approach, in this general context, relies on our use of Gaussian Hilbert spaces, and of associated sequences of independent, identically distributed i.
But this then further introduces a host of choices, and of these we identify one which is universal in a sense made precise in Sections In this paper, we will focus on Gaussian stochastic processes, but we also offer applications of our results to certain random functions Sec- tion 9 which involve non-Gaussian distributions.
Similarly, a host of simulation approaches involve non-Gaussian choices. The purpose of the paper is three-fold. First we study i a univer- sal choice of sample space for a family of L2 Gaussian noise processes. While these processes have appeared in one form or the other in prior literature, the choice of sample spaces has not been studied in a way that facilitates comparisons.
We index these Gaussian noise processes by the set of regular measures in some fixed measure space M, B , with B some given Borel sigma-algebra of subsets in M. Secondly we make precise ii equivalence in this category of Gaussian noise processes, and we prove a uniqueness theorem, where uniqueness is specified by a specific measure isomorphism of the respective sample spaces. In our third result iii , given a fixed measure space M, B , we identify a Hilbert space H , with the property that the Gaussian noise process indexed by H universal envelope of all the Gaussian noise processes from ii.
For readers not familiar with Gaussian processes, for the present pur- pose, the following are helpful: [3, 6, 9, 19, 22, 23, 24, 45, 46]; for infi- nite products and applications, see [34], [49], and [4], [7].
The universal Hilbert space from iii is used in a different context [11, 22, 28, 37, 38]. For a small sample of recent applications, we cite [16, 21, 27] For re- producing kernel Hilbert spaces, see, for example, [2, 44]. In the way of presentation, it will be convenient to begin with a quick review of infinite products, this much inspired by the pioneering paper [34] by Kakutani.
Preliminaries Below we present a framework of Gaussian Hilbert spaces. These in turn play a crucial role in the study of positive semi-definite kernels, and their associated reproducing kernel Hilbert spaces, see Sections 9- In its most general form, the theory of Gaussian Hilbert spaces H is somewhat abstract, and it is therefore of interest, for particular cases of H, to study natural decompositions into cyclic components in H which arise in applications, and admit computation.
Hence we begin with those processes whose covariance function may be deter- mined by a fixed measure. Even this simpler case generalizes a host of Gaussian processes studied earlier with the use of Gelfand triples built over the standard Hilbert space L2 Rd , dx , with dx denoting the Lebesgue measure, with the use of Laurent Schwartz theory of tem- pered distributions.
Our present framework is not confined to the Eu- clidean case. Indeed, in applications to measurement, in physics, and in statistics, it is often not possible to pin down a vari- able as a function of points in the underlying space M. As a result, it has proved useful to study processes indexed by sigma-algebras of subsets of M. In our consideration of random variables, of Hilbert spaces, and of Gaussian stochastic processes, it will be convenient for us to restrict to the case of real-valued functions and real Hilbert spaces.
One instance when complex Hilbert spaces are needed is the introduction of Fourier bases, i. The restricting assumption is sigma-finiteness, i. Definition 2. The finite dimensional case is dealt with separately. Remark 2. With the specifications in Definitions 2. In the case of Definition 2. Before getting to this, we must prepare the ground with some technical tools. This is the purpose of the next section on infinite products, and discrete Gelfand-triples. To do this, we will be introducing a suitable Gelfand triple see 3.
As the data in 3. Further note that the sets in 3. Lemma 3. Proof: We begin with the assertion 3. We now prove the assertion 3. With 3. The proof of 3. Now 3. Then 3. Indeed, the function 3. We need to prove the second claim in 3. Assume the contrary, i. Corollary 3. Let M, B be as in Theorem 3. Then, in the representation 3. Now use 3. N 0, 1 variables. A Hilbert space of sigma-functions In spectral theory, in representation theory see e.
Naturally, a given practical problem may not by itself entail a Hilbert space, and, as a result, one must be built by use of the inherent geometric features of the problem. In these applications it has proved useful to build the Hilbert space from a set of equivalence classes. It turns out see [38] that the set of such equivalence classes acquire the structure of a Hilbert space, called a sigma-Hilbert space. Further we show through applications Sections 7 and 8 that these sigma-Hilbert spaces form a versatile tool in the study of Gaussian processes.
These Gaussian processes are indexed by a choice of a suitable sigma-algebras of subsets of M. Definition 4. It is known see [38] , that 4. The Gaussian processes from Definition 2. Consider M, B as in Definition 4. Then, 4. Then, r! For the justification of 4. As a result we are able to show Theorem 5.
Definition 5. Let M, B be fixed, and let H denote the corre- sponding Hilbert space of sigma-functions; see Definition 4. Then the map p 5. We claim that it is onto. Let H be the sigma-Hilbert space of Definition 4. Let M, B be as in Section 2. Then, the map p 5. For the sum in H we have equation We now turn to the linearity of W 4. Hence, r r! Then, in view of 4. This is spelled out in Theorems 6.
N 0, 1 random variables. Theorem 6. We proceed in a number of steps. To see this, we use the construction in Proposition 4. The rest of the assertion is clear. STEP 2: We show that the sum on the right-hand-side of 6. Hence, the contributions to the two sides in 6. One involves a renormalization, somewhat subtle.
We now explain the connections between the present construction and the processes we built in [6, 5]. Application 6. As a corollary we have: Corollary 6. For some recent work on the fractional Brownian motion, see also [1, 8, 32, 36]. This we do in Theorem 7. Theorem 7. N 0, 1 family, and 6. Using again 6. Set 7. N 0, 1 system from Theo- rem 6. Note that iii in the Corollary says that 7. Now, formula 6. This proves 7.
Then, A is a Markov operator see [10] , i. Proof: Note that in ii and iii the symbol 1 denote the constant function equal to 1 in the respective measured spaces.
Properties i and ii are clear. We also make use of Theorem 7. See also [11]. Substitution of 7. See [12] for the latter. Iterated function systems The purpose of the present section is to give an application of the theorems from Sections 6 and 7 to iterated function systems IFS , see e. Such IFSs arise in geometric measure theory, in har- monic analysis, and in the study of dynamics of iterated substitutions with rational functions on Riemann surfaces ; hence the name iter- ated function system.
With an IFS, we have the initial measure space M and a Borel sigma-algebra B, coming with an additional structure, a system of measurable endomorphisms. For background, see e. Given a measure space M, B as in Section 2, i. This is in particular the case in applications to Riemann surfaces, where R is typically a rational function. Definition 8. Remark 8. Special cases of IF S have been widely studied in the literature; see e.
Lemma 8. Proof: In principle there are issues with passing the transformation onto equivalence classes, but this can be dome via an application of Lemma 4. Hence in studying 8. Theorem 8. We are now ready to verify 8. In this computation we make use of 8. Gaussian versus non-Gaussian In this section we show that the theory, developed above, initially for Gaussian Hilbert spaces, applies to some non-Gaussian cases; for ex- ample to those arising in the study of random functions.
To make this point specific, we address such a problem for the special case of a concrete random power series, studied as a family of infinite Bernoulli convolutions on the real line. We know, see [40], that every positive definite function may be realized in a Gaussian Hilbert space. Definition 9. If T is a set, then the function 9. The following is an important example of a solution to the problem 9.
In its simplest form, it may be presented as follows: Proposition 9. For some of the fundamentals in the theory of Bernoulli convolutions, we refer to [29, 41, 42]. Then the distribu- tion 9. Set 9. Remark 9. Otherwise it is absolutely continuous on a subset in [ 12 , 1 of full measure. Corollary 9. In this case, we may take 1 9. Proof: Using 9. This fact is isolated in the corollary below. For the literature on this we cite [13, 14, 15, 50].
Us- ing 9. It is our aim in Theorem 9. One is in particular interested in 9. Lemma 9. Let H be the reproducing kernel Hilbert space from 9. Proof: The conclusion follows from the basic axioms of reproducing kernel Hilbert spaces once we verify that Z 9.
The reproducing kernel Hilbert space H from 9. Indeed, since J is isometric, ran J is closed. About 9. As for estimating 9. Under the isometry in 9. Boundaries of positive definite functions In this section we apply our results from Sections 3 and 7 into a gen- eral boundary analysis for an arbitrarily given non-degenerate posi- tive definition function Definition 9.
For example T may represent the vertices in some infinite graph, and C may be some associated energy form of the graph G, induced by an electric network of G; see e. A second recent application of reproducing kernels and their RKHSs, is the theory of supervised learning; see e. The problem there is a prediction of outputs based on observed samples; and for this the kernel enters in representations of samples.
As example of this is the Hardy space H2 D see Defi- nition 9. In this general case, the aim is to provide a Gaussian measure space associated to an arbitrary given positive definite function This measure space will be denoted by bdrC T , and it should be offer a direct integral representation for Uniform A pair of impulses Gaussian Rayleigh. Here, its Fourier transform is also coming a Gaussian pulse shape. So, the amplitude spectrum is the same as the Gaussian pulse. Get Started for Free Download App.
Find the output autocorrelation R YY 0. Answer Detailed Solution Below 3. U and V are two independent zero mean and Gaussian. Answer Detailed Solution Below Option 2 : 1. Answer Detailed Solution Below 0.
Process become Random variable.
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