Abstract and Applied Analysis

Volume 2014 (2014), Article ID 457051, 7 pages

http://dx.doi.org/10.1155/2014/457051

Research Article

Karhunen-Loève Expansion for the Second Order Detrended Brownian Motion

Yongchun Zhou,¹ Xiaohui Ai,² Minghao Lv,¹ and Boping Tian¹

¹Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

²Department of Mathematics, Northeast Forestry University, Harbin 150040, China

Received 8 March 2014; Revised 29 May 2014; Accepted 28 June 2014; Published 16 July 2014

Academic Editor: Salvador Romaguera

Copyright © 2014 Yongchun Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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).

References

P. Deheuvels, G. Peccati, and M. Yor, “On quadratic functionals of the Brownian sheet and related processes,” Stochastic Processes and Their Applications, vol. 116, no. 3, pp. 493–538, 2006.

P. Deheuvels, “A Karhunen-Loève expansion for a mean-centered Brownian bridge,” Statistics and Probability Letters, vol. 77, no. 12, pp. 1190–1200, 2007.

P. Deheuvels and G. V. Martynov, “Karhunen-Loève expansions for weighted Wiener processes and Brownian Bridges via Bessel Functions,” Progress in Probability, vol. 55, pp. 57–93, 2003.

P. Deheuvels, “Karhunen-Loève expansions of mean-centered Wiener processes,” High Dimensional Probability, vol. 51, pp. 62–76, 2006.

X. Ai, W. V. Li, and G. Liu, “Karhunen-Loeve expansions for the detrended Brownian motion,” Statistics & Probability Letters, vol. 82, no. 7, pp. 1235–1241, 2012.

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, UK, 1952.

I. B. MacNeill, “Properties of sequences of partial sums of polynomial regression residuals with applications to tests for change of regression at unknown times,” The Annals of Statistics, vol. 6, no. 2, pp. 422–433, 1978.

G. V. Martynov, “A generalization of Smirnov’s formula for the distribution functions of quadratic forms,” Theory of Probability & Its Applications, vol. 22, no. 3, pp. 614–620, 1977.

N. Smirnov, “Table for estimating the goodness of fit of empirical distributions,” Annals of Mathematical Statistics, vol. 19, pp. 279–281, 1948.

M. Barczy and E. Iglói, “Karhunen-Loève expansions of α-Wiener bridges,” Central European Journal of Mathematics, vol. 9, no. 1, pp. 65–84, 2011.

W. V. Li, “Comparison results for the lower tail of Gaussian seminorms,” Journal of Theoretical Probability, vol. 5, no. 1, pp. 1–31, 1992.

W. V. Li, “Limit theorems for the square integral of Brownian motion and its increments,” Stochastic Processes and Their Applications, vol. 41, no. 2, pp. 223–239, 1992.