# Stationary Stochastic Processes. (MN-8)

TAKEYUKI HIDA
Pages: 168
https://www.jstor.org/stable/j.ctt13x0tn0

1. Front Matter
(pp. None)
2. Preface
(pp. i-i)
(pp. ii-vi)
4. §.0. Introduction
(pp. 1-3)

The ideas presented in this course were inspired by certain investigations of stationary stochastic processes using nonlinear operators acting on them, e.g. nonlinear prediction theory and the discussions about innovations for a stationary process.

We shall be interested in functionals of certain basic or fundamental stationary stochastic processes. In the discrete parameter case, they are the familiar sequences of independent identically distributed random variables. In the continuous parameter case, which concerns us, they are stationary processes with independent values at every moment in the sense of Gelfand. A particularly important role will be played by the so-called (Gaussian) white noise...

5. §.1. Background
(pp. 3-13)

In this article we shall prepare some basic concepts as background for our discussions.

Let Ω be a certain non-empty set. Each element ω of Ω is supposed to be an elementary event or a probability parameter. Let Ⓑ be a σ-field of subsets of Ω, i.e., Ⓑ satisfies

i) If Ⓑ∋Anfor n = 1,2,… , then$\underset{\text{n}}{\mathop{\cup }}\,$An∈Ⓑ

ii) If Ⓑ∋A, then Ac≡ Ω − A ∈ Ⓑ

iii) Ⓑ∋ϕ (empty set).

A set belonging to Ⓑ is called an event or a measurable set (w. r. t. Ⓑ). A (countably additive) measure P defined on Ⓑ...

6. §.2. Brownian motion.
(pp. 14-30)

First we shall define a standard Brownian motion and give one possible construction for it. Then we shall see that Brownian motion determines a measure on function space.

A system of r.v.'s B(t,ω), ω ∈ Ω, 0 ≤ t < ∞, is called a standard Brownian motion if it satisfies the following conditions :

i) The system is {B(t); 0 ≤ t < ∞} is Gaussian

ii) B(0) = 0, a.s.

iii) The probability distribution of is B(t) − B(s) is N(0, |t−s|).

From iii) it follows that E(B(t)) = 0 for every t. Also for u < s < t, the relation

V(B(t)−B(u))...

(pp. 31-61)

We shall begin with very simple and elementary examples of additive processes, i.e. the Poisson process and compound Poisson processes, the sample functions of which are quite different from those of Brownian motion. Then we shall discuss, as a generalization of compound Poisson processes, Lévy processes with stationary increments.

As in the case of Brownian motion, a Lévy process determines a probability measure on function space. The tension group acting on the function space will serve to characterize stable processes. This leads to a probabilistic interpretation of Bochner's famous theory of subordination.

Let ξn(ω), ω ∈ Ω(Ⓑ,P), n=1,2,…, be a...

8. §4. Stationary processes
(pp. 62-85)

This article starts out by discussing probability measures on the space of (generalized) functions, as well as the need for such measures. This leads to our defining a stationary process in the generalized sense, as a probability measure μ on function space which is invariant under any shift of the argument of the functions. For any such measure μ we have the Hilbert space L2(μ) of functionals of sample functions of the given stationary process.

We do some analysis on the space L2(μ) in section 4.4. Finally we consider the very important class of stationary process with independent value at...

9. § 5 Gaussian processes.
(pp. 86-93)

We shall first give a definition of Gaussian process. The class of Gaussian processes is one of the important classes of (generalized) stochastic processes. Then we shall deal with linear operations acting on them. The discussions there will show that the Gaussian white noise plays a dominant role in the study of Gaussian processes.

Throughout this section the nuclear space E will be assumed to be either the space ℒ or the space ℋ (for definitions we refer to Examples i) and ii) in § 4.2) in order to simplify our discussions.

Let (E*, μ) be a generalized stochastic process...

10. § 6. Hilbert space (L2) arising from white noise.
(pp. 94-105)

Let E be a real nuclear space such that$\text{E}\subset {\cal{H}}={{\text{L}}^{2}}({{\text{R}}^{\text{l}}},\text{Leb})\subset \text{E}^{*}$and such that E is stable under the shifts St, − ∞ < t < ∞. The Gaussian white noise (abbr. W. N.) is a stochastic process with characteristic functional$\text{C}(\xi)=\exp \{-\frac{1}{2}\int{\xi{{(\text{t})}^{2}}\text{dt}}\},\ \xi\in \text{E}.$

Since C(ξ) is St-invariant, W.N. is a stationary process. As was seen in the example of § 5.2., W.N. has independent values at every moment. Furthermore we know that is a Gaussian process and in particular the system { ; ξ ∈ E} is a Gaussian system.

We now proceed to the analysis of (L2) = L2(E*, μ).

Every assertion...

11. §7. FLow of the Brownian motion.
(pp. 106-112)

Let μ on (E*, Ⓑ) be W. N. (white noise) with the characteristic functional$\text{C}(\xi)=\exp \{-\frac{1}{2}\int{\xi{{(\text{t})}^{2}}\text{dt}}\},\ \xi\in \text{E}.$

Since W. N. is stationary, {Ttt real} is a flow on (E*, μ). According to the discussion in §4.3 the Utdefined by

Utφ(x) = φ(Ttx) , φ ∈ (L2),

form a strongly continuous one parameter group of unitary operators acting on (L2). Therefore we can appeal to Stone's theorem which asserts that {Ut} has a spectral decomposition:

(1) Ut= ∫ eitλdE(λ),

where {E(λ); λ real} is a resolution of the identity.

We are interested in the spectral type of {Ut} (or {Tt})...

12. §8. Infinite dimensional rotation group
(pp. 113-119)

After H. Yosizawa we consider the collection 0(E) of all linear transformations {g} on E satisfying the following two conditions:

i) g is a homeomorphism on E,

(1)

ii) ∫(gξ) (t)2dt = ∫ξ(t)2dt,

where E is a nuclear space dense in L2(Rl) Obviously the set 0(E) of all such g's forms a group with respect to the operation (g1g2)ξ = g1(g2ξ). The group 0(E) is called the infinite dimensional rotation group. Since the characteristic functional of W.N. is Invariant under 0(E), the measure μ of W.N. is invariant under g* the adjoint of g, g ∈ 0(E).

We first investigate...

13. §9. Fourier Analysis on (L2), motion group and Laplacian.
(pp. 120-134)

We have already discussed the transformation τ. This has a formal resemblance to the ordinary Fourier transform, but it differs 2 crucially in that it maps the (L2) not onto itself but to the reproducing kernel Hilbert space${\cal {F}}({\text {{E},{C}}})$. We first discuss another k of transformation which does play the role of Fourier transform on (L2). We then proceed to the infinite dimensional motion group and observe the intimate relation with this Fourier transform. Finally we introduce the infinite dimensional Laplacian, which is discussed in connection with 0and the direct sum decomposition of (L2).

Let φ(x) be in...

14. §10. Applications.
(pp. 135-146)

This section will be devoted, to brief remarks on several 2 applications of our analysis on the space (L).

(On the fourth anniversary of N. Wiener's death)

We are going to discuss a stationary stochastic process obtained through a nonlinear network from a Brownian input. We refer to N. Wiener [19, Lecture 10] for a discussion of why a Brownian input is fitting for the analysis of networks. It is quite reasonable to assume that the given network is nonexplosive, deadbeat, and so forth. Ihe output through the network is a functional of the Brownian motion which is the input....

15. §.11. Generalized White Noise.
(pp. 147-162)

Up to now we have been concerned with Gaussian white noise, which was, roughly speaking, the derivative of Brownian motion. We shall generalize it to the derivative of a Levy process with stationary increments, that is, a generalized white noise.

Let E he a nuclear space such that

The generalized white noise is a stochastic process the characteristic functional of which is given by

In particular, if is expressed in the form

the corresponding process is called the Poisson white noise with jump u.

Observing the expression (2) we understand that a generalized white noise is composed of three kinds...

16. Appendix
(pp. A-1-A-4)