Karhunen-Loève Expansion for the Second Order Detrended Brownian Motion

Abstract

Based on the norm in the Hilbert Space , the second order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion into the space spanned by nonlinear function subspace. Karhunen-Loève expansion for this process is obtained together with the relationship of that of a generalized Brownian bridge. As applications, Laplace transform, large deviation, and small deviation are given.

1. Introduction

Let be a centered and continuous Gaussian process on with covariance function

The Karhunen-Loève expansion of is given by the (convergent in mean squares) series where is a sequence of i.i.d. random variables and is at most the countable set of eigenvalues of Fredholm integral operator and forms an orthogonal sequence in and .

Deheuvels et al. in [1–4] provided the Karhunen-Loève expansions for the processes that are related with Brownian motion. The Karhunen-Loève expansion for detrended Brownian motion has been studied by Ai et al. [5]. Note that the detrended Brownian motion in [5] can be viewed as projection to a constant function subspace in . That is,

To generalize the projection idea into nonlinear detrended process, now we consider and the optimal constant satisfy

It is easy to obtain

Let we have

Now we can define the second order detrended process

2. Main Results

We give the following lemma that provides the explicit covariance function.

Lemma 1. For convenience, we add into formula (11), that is where is given in (8).

Proof. Consider and is a mean zero Gaussian process; we obtain We notice that Substituting (16), (17), and (19) into (15), we derive

Lemma 2 (see [3]). If , , , then the condition is equivalent to the identity

In the following, we will give some preliminaries, notions, and facts that are needed in Theorem 3. For , is Bessel function [6] with index and the positive zeros of are infinite sequence . When , , the positive zeros of , are , , , and they are in such a way that

Now we can state one of the main results of this paper.

Theorem 3. For the second order detrended Brownian motion and a generalized Brownian bridge with in [7],
One has the distribution identities where and denote two independent sequences of independently and identically distributed random variables.

Proof. By straightforward induction based on the equation and splitting the integration range from , we get By differentiation of both sides of (23) with respect to , we have By differentiation of both sides of (24) with respect to , we have We can simplify this equation to where
We solve the inhomogeneous second differential equation to obtain
We substitute into (28) and (29) to obtain
In order that there are nonzero choices for , the determinant of the above two equations has to be zero, which can be written as where
We obtain, after some simplification,
Then is an eigenvalue if and only if (34) holds. We therefore obtain with .
According to the trigonometric function formula
we can observe that where , are Bessel functions as follows: which gives two sequences of eigenvalues of (37), namely, and .
Similarly, we can obtain the two eigenvalues , corresponding to those of integral operator of a generalized Brownian bridge . Note that the integral operator is Actually, in Lemma 2, we have the distribution identities

Remark 4. From (11) and (22), we derive that by using the Rayleigh’s formula, for and (see, e.g., [3, (1.91), page 77] and [6, page 502]).
To check (41), from (11), we infer that which is in agreement with (41).

3. Applications

In this section, the relevant applications of Karhunen-Loève expansion are given.

Proposition 5. For each , one has

Proof. where and .

Proposition 6. If , then where , .

Proof. It can be proved by the Smirnov formula [8, 9], formula (23), and the definition of the Fredholm determinant. Similar proof method can be found from Proposition 3.3 in [10].

Next, we give the large deviation and small deviation probabilities of the second order detrended Brownian motion with respect to the norm in the Hilbert Space .

Proposition 7. Consider ,

Proof. By Deheuvels [2] and Martynov [8], we have for all we take and into (47), and then the proof is completed.

Proposition 8. There exists a constant such that

Proof. We start with proving (48) by recalling Li, 1992 [11, 12].
Given two sequences and with we have, as ,
By the asymptotic formula for zeros of Bessel function then , , and , , which satisfy (49) and by the distribution identity and (50), there exists a constant , such that
Also, for all , there exists a constant , such that, as , Connecting (52) with (53), we can obtain the proposition.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Fund (71350005), Heilong Jiang Province Natural Science Fund (G200815), and the Fundamental Research Funds for the Central Universities (no. DL13BBX10).