Uniform convergence of wavelet expansionsof Gaussian random processes Short title: Uniform convergence of wavelet expansions

Uniform convergence of wavelet expansions
of Gaussian random processes
Short title: Uniform convergence of wavelet expansions

Yuriy Kozachenko ykoz@ukr.net Andriy Olenko a.olenko@latrobe.edu.au Olga Polosmak DidenkoOlga@yandex.ru Department of Probability Theory, Statistics and Actuarial Mathematics, Kyiv University, Kyiv, Ukraine Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia
 This is an Author’s Accepted Manuscript of an article published in the Stochastic Analysis and Applications, Vol. 29, No. 2, 169–184. [copyright Taylor & Francis], available online at: http://www.tandfonline.com/ [DOI:10.1080/07362994.2011.532034]
Abstract

New results on uniform convergence in probability for the most general classes of wavelet expansions of stationary Gaussian random processes are given.

keywords:
Convergence in probability, Gaussian process, Random process, Uniform convergence, Wavelets
Msc:
60G10, 60G15, 42C40
journal: Stochastic Analysis and Applications\newdefinition

rmkRemark

1 Introduction

In various applications in data compression, signal processing and simulation, it could be useful to convert the problem of analyzing a continuous-time random process to that of analyzing a random sequence, which is much simpler. Multiresolution analysis provides an efficient framework for the decomposition of random processes. This approach is widely used in statistics to estimate a curve given observations of the curve plus some noise.

Various extensions of the standard statistical methodology were proposed recently. These include curve estimation in the presence of correlated noise. For these purposes the wavelet based expansions have numerous advantages over Fourier series, see kur (); wal (), and often lead to stable computations, see pho ().

However, in many cases numerical simulation results need to be confirmed by theoretical analysis. Recently, a considerable attention was given to the properties of the wavelet transform and of the wavelet orthonormal series representation of random processes. More information on convergence of wavelet expansions of random processes in various spaces, references and numerous applications can be found in cam (); did (); ist (); kozvas (); kozpol (); kur (); zha ().

We focus out attention on uniform convergence of wavelet expansions for stationary Gaussian random processes. We consider a random process such that for all .

In the book har () wavelet expansions of functions bounded on were studied in different spaces. Obtained results were applied by several authors to investigate wavelet expansions of random processes bounded on . However the majority of random processes, which are interesting from theoretical and practical application points of view, has almost surely unbounded sample paths on . In numerous cases developed deterministic methods may not be appropriate to investigate wavelet expansions of stochastic processes. It indicates the necessity of elaborating special stochastic techniques.

In the paper we consider stationary Gaussian random processes and their approximations by sums of wavelet functions

(1)

where

Contrary to many theoretical results (see, for example, kozvas (); kur ()) with infinite series form of in direct numerical implementations we always consider truncated series like (1), where the number of terms in the sums is finite by application reasons. However, there are almost no stochastic results on uniform convergence of finite wavelet expansions to

We show that, under suitable conditions, the sequence converges in probability in Banach space , i.e.

when and for all More details on the general theory of random processes in the space can be found in bulkoz ().

The numbers and of terms in the truncated series can approach infinity in any arbitrary way. Thought about this way, one sees that the paper deals with the most general class of such wavelet expansions in comparison with particular cases considered by different authors, see, for example, cam (); kur ().

Most known results (see, for example, cam (); did (); ist (); wong (); zha ()) concern the mean-square convergence, but for practical applications one needs to require uniform convergence.

We present the first result on stochastic uniform convergence of general finite wavelet expansions in the open literature. It should be mentioned that in the general case the random coefficients in (1) may be dependent and form an overcomplete system of basis functions. That is why the implementation of the proposed method has promising potential for nonstationary random processes.

The organization of this article is the following. In the second section we introduce the necessary background from wavelet theory and certain sufficient conditions for mean-square convergence of wavelet expansions in the space obtained in kozpol (). In §3 we formulate and discuss the main theorem on uniform convergence in probability of the wavelet expansions of stationary Gaussian random processes. The next section contains the proof of the theorem. Conclusions are made in section 5.

2 Wavelet representation of random processes

Let be a function from the space such that and is continuous at where

is the Fourier transform of

Suppose that the following assumption holds true:

There exists a function , such that has the period and

In this case the function is called the -wavelet.

Let be the inverse Fourier transform of the function

Then the function

is called the -wavelet.

Let

(2)

It is known that the family of functions is an orthonormal basis in (see, for example, chu (); dau ()).

An arbitrary function can be represented in the form

(3)

The representation (3) is called a wavelet representation.

The series (3) converges in the space i.e.

The integrals and may also exist for functions from and other function spaces. Therefore it is possible to obtain the representation (3) for function classes which are wider than .

Let be a standard probability space. Let be a random process such that It is possible to obtain representations like (3) for random processes, if sample trajectories of these processes are in the space However the majority of random processes do not possess this property. For example, sample paths of stationary processes are not in the space (a.s.).

We want to construct a representation of the kind (3) for with mean-square integrals

Consider the approximants of defined by (1). Theorem 1 below guarantees the mean-square convergence of to

Assumption S. har () For the -wavelet there exists a function such that , is a decreasing function, (a.e.) and

Let denote a non decreasing even function on with

Theorem 1

kozpol () Let be a random process such that for all and its covariance function is continuous. Let the -wavelet and the -wavelet be continuous functions and the assumption S hold true for both and Suppose that there exists a function and such that for all

If

then

  1. in mean square when and for all

{rmk}

For stationary Gaussian processes we can choose and 

3 Uniform convergence of wavelet expansions for Gaussian random processes

Theorem 2

kozsli () Let , . Let be a sequence of Gaussian stochastic processes. Assume that all are separable in and

where is a monotone increasing function such that when

Suppose that for some

(4)

where is the inverse function of . If the processes converges in probability to the process for all , then converges in probability to in the space

{rmk}

For example, it is easy to check that the assumption (4) holds true for

when

Now we are ready to formulate the main result.

Theorem 3

Let be a stationary separable centered Gaussian random process such that its covariance function is continuous. Let the -wavelet and the corresponding -wavelet be continuous functions and the assumption S hold true for both and Suppose that the following conditions hold:

  1. there exist and

  2. and when

  3. there exist and such that

  4. there exists and

  5. and for

Then uniformly in probability on each interval when and for all

Before seeing the proof of the theorem, we clarify the role of some of the assumptions. Two kinds of assumptions were made:

  • conditions 1-4 on the wavelet basis and

  • conditions 5 and 6 on the random process.

Contrary to many other results in literature, our assumptions are very simple and can be easily verified.

Conditions 1-4 are related to the smoothness and the decay rate of the wavelet basis functions and It is easy to check that numerous wavelets satisfy these conditions, for example, the well known Daubechies, Battle-Lemarie and Meyer wavelet bases. Conditions 5 and 6 on the random process are formulated in terms of the spectral density These conditions are related to the behavior of the high-frequency part of the spectrum. Both sets of assumptions are standard in the convergence studies.

If we narrow our general class of wavelet expansions and impose some additional constraints on rates of the sequences we can enlarge classes of wavelets bases and random processes in the theorem. It will be seen from the proof of the theorem that all we need are conditions which guarantee that the series and are convergent.

4 Proof of the main theorem

From conditions 2 and 4, it follows that

for any Thus to prove the theorem we only consider the case

We first prove that for some the inequality

(5)

holds true for all and

By (1) we obtain

We will show how to handle , then similar techniques can be used to deal with the remaining term

satisfies the inequality

Let us consider By means of Parseval’s theorem we deduce

The order of integration can be changed because

The last expression is finite due to (2), the assumption S, the estimate (14), and the representation

(6)

We begin with the case

Applying integration by parts, the assumptions of the theorem, and twice the intermediate value theorem of derivatives yields the following

where

because of conditions 5 and 6.

We use (6) to estimate the term Then

Therefore

(7)

By the inequality (59) given in kozroz ()

(8)

An application of this inequality to the second integral in (7) results in

where

In the following derivations, we will use conditions 1, 4 and the estimates

(9)

where

The integral is finite because of the boundedness of and condition 4.

Using these facts, we get

(10)

Similarly, we can estimate the first integral. It is easy to see that

Note that, by (8) and (9):

(11)

Due to (koztur, , Lemma 4.2)

(12)

where depends only on and

Applying inequalities (11), (12) and we get

Using above inequalities the first integral can be estimated as follows:

(13)

where

The integrals and are finite because is bounded:

(14)

Using (10) and (13), we obtain:

(15)

where

Thus

(16)

where

Using the inequality