Uniform convergence of wavelet expansions
of Gaussian random processes
Short title: Uniform convergence of wavelet expansions
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
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
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 ().
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.
An arbitrary function can be represented in the form
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
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
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
in mean square when and for all
For stationary Gaussian processes we can choose and
3 Uniform convergence of wavelet expansions for Gaussian random processes
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
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
For example, it is easy to check that the assumption (4) holds true for
Now we are ready to formulate the main result.
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:
there exist and
there exist and such that
there exists and
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:
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
for any Thus to prove the theorem we only consider the case
We first prove that for some the inequality
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
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
We use (6) to estimate the term Then
By the inequality (59) given in kozroz ()
An application of this inequality to the second integral in (7) results in
The integral is finite because of the boundedness of and condition 4.
Using these facts, we get
Similarly, we can estimate the first integral. It is easy to see that
where depends only on and
Using above inequalities the first integral can be estimated as follows:
The integrals and are finite because is bounded:
Using the inequality