# Karhunen-Loève Expansion for the Second Order Detrended Brownian Motion

Authors: Yongchun Zhou; Xiaohui Ai; Minghao Lv; Boping Tian
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Based on the norm in the Hilbert SpaceL2[0,1], 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.


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## Body

## 1. Introduction

LetX={X(t),0≤t≤1} be a centered and continuous Gaussian process on [0,1] with covariance function
(1)KX(t,s)=EX(t)X(s).The Karhunen-Loève expansion ofX is given by the (convergent in mean squares) series
(2)X(t)=∑k=1∞ηkλkfk(t),
where {ηk,k≥1} is a sequence of i.i.d. N(0,1) random variables and {λk,k≥1} is at most the countable set of eigenvalues of Fredholm integral operator
(3)TXf(t)=∫01KX(t,s)f(s)ds{fk(t),k≥1} and forms an orthogonal sequence in L2[0,1] and ∫01KX(t,t)dt<∞.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 L2[0,1]. That is,
(4)∫01W^1(t)2dt=min⁡c1,c2∫01(W(t)-c1-c2t)2dt.To generalize the projection idea into nonlinear detrended process, now we consider(5)min⁡c1,c2,c3∫01(W(t)-c1-c2t-c3t2)2dt
and the optimal constant cj satisfy
(6)∂∂cj∫01(W(t)-c1-c2t-c3t2)2dt=0,j=1,2,3.It is easy to obtain(7)c1=9∫01W(s)ds-36∫01W(s)sds+30∫01W(s)s2ds,c2=-36∫01W(s)ds+192∫01W(s)sds-180∫01W(s)s2ds,c3=30∫01W(s)ds-180∫01W(s)sds+180∫01W(s)s2ds.Let(8)A=(aij)3×3=(9-3630-36192-18030-180180);
we have
(9)cj=∑i=13aij∫01si-1W(s)ds,j=1,2,3.Now we can define the second order detrended process(10)W^2(t)=W(t)-∑j=13cjtj-1=W(t)+(-9+36t-30t2)∫01W(s)ds+(36-192t+180t2)∫01W(s)sds+(-30+180t-180t2)∫01W(s)s2ds.


## 2. Main Results

We give the following lemma that provides the explicit covariance function.Lemma 1.
For convenience, we addKX(s,t) into formula (11), that is
(11)KX(s,t)=E(W^2(t)W^2(s))=t∧s-∑p,q=13apq(tp-tp+1p(p+1))sq-1-∑i,j=13aij(si-si+1i(i+1))tj-1+∑p,q=13∑i,j=13aijapqp+i+2(p+1)(p+i+1)(i+1)tj-1sq-1,
where aij,apq,i,j,p,q=1,2,3 is given in (8).Proof.
Consider(12)W^2(t)=W(t)-∑j=13cjtj-1,0≤t≤1
and W^2(t) is a mean zero Gaussian process; we obtain
(13)E(W^2(t)W^2(s))=EW^2(t)W^2(s)=E(W(t)-∑j=13cjtj-1)(W(s)-∑q=13cqsq-1)=E(W(t)-∑i,j=13aij(∫01ui-1W(u)du)tj-1)·E(W(s)-∑p,q=13apq(∫01vp-1W(v)dv)sq-1).
We notice that
(14)E(W(t)W(s))=t∧s,(15)E(W(t)∫01vp-1W(v)dv)=E(∫01W(t)W(v)vp-1dv)=∫01(t∧v)vp-1dv=∫0tvpdv+∫t1tvp-1dv=tp-tp+1p(p+1),(16)E(∫01ui-1W(u)du)(∫01vp-1W(v)dv)=∫01ui-1E(W(u)∫01vp-1W(v)dv)du=∫01ui-1(up-up+1p(p+1))du=p+i+2(p+1)(p+i+1)(i+1).
Substituting (16), (17), and (19) into (15), we derive
(17)E(W^2(t)W^2(s))=t∧s-∑p,q=13apq(tp-tp+1p(p+1))sq-1-∑i,j=13aij(si-si+1i(i+1))tj-1+∑p,q=13∑i,j=13aijapqp+i+2(p+1)(p+i+1)(i+1)tj-1sq-1.Lemma 2 (see [3]).
Ift∈[0,1], ςj(t)=∑k=1∞ωkλk,jek,j(t), j=1,2,…, then the condition
(18)∫[0,1]ς12(t)dt=law∫[0,1]ς22(t)dt
is equivalent to the identity
(19)λk,1=λk,2∀k≥1.In the following, we will give some preliminaries, notions, and facts that are needed in Theorem3. For v>-1, Jv(·) is Bessel function [6] with index v and the positive zeros of Jv(·) are infinite sequence 0<zv,1<zv,2<⋯. When v=3/2, v=5/2, the positive zeros of J3/2,k, J5/2,k are z3/2,k, z5/2,k, k=1,2,…, and they are in such a way that
(20)0<z3/2,1<z5/2,1<z3/2,2<⋯.Now we can state one of the main results of this paper.Theorem 3.
For the second order detrended Brownian motionW^2(t) and a generalized Brownian bridge B2(t) with n=2 in [7],
(21)B2(t)=B(t)-136t(60t2+18t-67)B(1)-t(60t2-96t+11)∫01B(s)ds+10t(12t2-18t+1)∫01B(s)sds.
One has the distribution identities(22)∫01W^2(t)2dt=law∫01B2(t)2dt=law∑k≥1ηk24z3/2,k2+∑k≥1ηk*24z5/2,k2,
where {ηk,k≥1} and {ηk*,k≥1} denote two independent sequences of independently and identically distributed N(0,1) random variables.Proof.
By straightforward induction based on the equation and splitting the integration range fromt, we get
(23)λf(t)=∫0tsf(s)ds+t∫t1f(s)ds-∑p,q=13apq(tp-tp+1p(p+1))∫01sq-1f(s)ds-∑i,j=13aijtj-1∫01(si-si+1i(i+1))f(s)ds+∑p,q=13∑i,j=13aijapqp+i+2(p+1)(p+i+1)(i+1)tj-1×∫01sq-1f(s)ds.
By differentiation of both sides of (23) with respect to t, we have
(24)λf′(t)=∫t1f(s)ds-∑p,q=13apq1-tpp∫01sq-1f(s)ds-∑i=1,j=23(j-1)aijtj-2∫01(si-si+1i(i+1))f(s)ds+∑p,q=13∑i=1,j=23aijapq(p+i+2)(j-1)(p+1)(p+i+1)(i+1)tj-2×∫01sq-1f(s)ds.
By differentiation of both sides of (24) with respect to t, we have
(25)λf′′(t)+f(t)=∑i,p,q=13ai3apq2(p+i+2)(p+1)(p+i+1)(i+1)∫01sq-1f(s)ds-2∑i=13ai3∫01(si-si+1i(i+1))f(s)ds+∑q=13a1q∫01sq-1f(s)ds+(∑q=13a2q∫01sq-1f(s)ds)t+(∑q=13a3q∫01sq-1f(s)ds)t2.
We can simplify this equation to
(26)λf′′(t)+f(t)+b1+b2t+b3t2=0,
where
(27)b1=-∑i,p,q=13ai3apq2(p+i+2)(p+1)(p+i+1)(i+1)∫01sq-1f(s)ds+2∑i=13ai3∫01(si-si+1i(i+1))f(s)ds-∑q=13a1q∫01sq-1f(s)ds,(28)b2=-∑q=13a2q∫01sq-1f(s)ds,(29)b3=-∑q=13a3q∫01sq-1f(s)ds.
We solve the inhomogeneous second differential equation to obtain(30)f(t)=c1cos⁡tλ+c2sintλ+2λb3-b1-b2t-b3t2.
We substitutef(t) into (28) and (29) to obtain
(31)(λsin1λ+6λcos⁡1λ-12λλsin1λ+6λ)c1+(-λcos⁡1λ+6λsin1λiiiiiiiiiiiiii+12λλcos⁡1λ-12λλ+λ)c2=0,(-2λsin1λ-14λcos⁡1λ+30λλsin1λ-16λ)c1+(2λcos⁡1λ-14λsin1λiiiiiiiiiiiiii-30λλcos⁡1λ+30λλ-3λ)c2=0.
In order that there are nonzero choices forc1,c2, the determinant of the above two equations has to be zero, which can be written as
(32)D11D22-D12D21=0,
where
(33)D11=λsin1λ+6λcos⁡1λ-12λλsin1λ+6λ,D12=-λcos⁡1λ+6λsin1λ+12λλcos⁡1λ-12λλ+λ,D21=-2λsin1λ-14λcos⁡1λ+30λλsin1λ-16λ,D22=2λcos⁡1λ-14λsin1λ-30λλcos⁡1λ+30λλ-3λ.
We obtain, after some simplification,(34)24λ2λ+4λλ=(24λ2-λ)sin1λ+(24λ2λ-8λλ)cos⁡1λ.
Thenλ≠0 is an eigenvalue if and only if (34) holds. We therefore obtain
(35)D(λ)=-720((24λ-7/2-λ-5/2)sinλ1/2iiiiiiiiiiii+(24λ-4-8λ-3)cos⁡λ1/2-24λ-4-4λ-3),
with D(0)=1.
According to the trigonometric function formula(36)sin1λ=2sin12λcos⁡12λ,cos⁡1λ=2cos⁡212λ-1=1-2sin212λ,
we can observe that(37)D11D22-D12D21=-12πλJ3/2(12λ)J5/2(12λ)=0,
where J3/2(z), J5/2(z) are Bessel functions as follows:
(38)J3/2(z)=2π·zπ(sinzz2-cos⁡zz),J5/2(z)=2π·zπ((-1z+3z3)sinz-3z2cos⁡z),
which gives two sequences of eigenvalues of (37), namely, (2z3/2,k)-2 and (2z5/2,k)-2.
Similarly, we can obtain the two eigenvalues(2z3/2,k)-2, (2z5/2,k)-2 corresponding to those of integral operator of a generalized Brownian bridge B2(t). Note that the integral operator is
(39)∫01K2(s,t)f(s)ds.
Actually, in Lemma 2, we have the distribution identities
(40)∫01W^2(t)2dt=law∫01B2(t)2dt=law∑k≥1ηk24z3/2,k2+∑k≥1ηk*24z5/2,k2.Remark 4.
From (11) and (22), we derive that
(41)∫01KX(t,t)dt=∫01E(W^2(t)2)dt=E∫01W^2(t)2dt=∑k≥114z3/2,k2+∑k≥114z5/2,k2=140+156=3140
by using the Rayleigh’s formula, for v=3/2 and v=5/2 (see, e.g., [3, (1.91), page 77] and [6, page 502]).
To check (41), from (11), we infer that
(42)∫01KX(t,t)dt=∫01[t-∑p,q=13apq(tqp-tp+qp(p+1))iiiiiiiiiiiiiiii-∑i,j=13aij(tji-ti+ji(i+1))iiiiiiiiiiiiiiii+∑p,q=13∑i,j=13aijapqiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∑p,q=13apq×p+i+2(p+1)(p+i+1)(i+1)tj+q]dt=3140
which is in agreement with (41).


## 3. Applications

In this section, the relevant applications of Karhunen-Loève expansion are given.Proposition 5.
For eachθ∈R, one has
(43)Eexp⁡(-θ22∫01W^2(t)2dt)={-720((24θ-7-θ-5)sinθ+(-24θ-8+8θ-6)cos⁡θiiiiiiiiiiiiiiiiiiiiiii+24θ-8+4θ-6)}-1/2.Proof.
(44)
E
exp
⁡
(
-
θ
2
2
∫
0
1
W
^
2
(
t
)
2
d
t
)
=
E
exp
⁡
(
-
θ
2
2
∑
k
=
1
∞
λ
k
ξ
k
2
)
=
∏
k
=
1
∞
(
1
+
λ
k
θ
2
)
-
1
/
2
=
(
D
(
-
θ
2
)
)
-
1
/
2
=
{
-
720
(
(
24
θ
-
7
-
θ
-
5
)
sin
θ
+
(
-
24
θ
-
8
+
8
θ
-
6
)
cos
⁡
θ
iiiiiiiiiiiiiiiiiiiiiii
+
24
θ
-
8
+
4
θ
-
6
)
}
-
1
/
2
,
where λ1>λ2>⋯>0 and ∑k=1∞λk<∞.Proposition 6.
Ifx>0, then
(45)P(∫01W^22(t)dt>x)=1π∑k=1∞(-1)k+1×∫γ2k-1γ2k(e-ux/2-24u-4-4u-3)|1/2)-1iiiiiiiiiiiiiiiiiiiiii×(-24u-4-4u-3)|)1/2)-1u(-24u-4-4u-3)|)1/2|-720((24u-7/2-u-5/2)sinu1/2-24u-4-4u-3)|1/2iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii+(24u-4-8u-3)cos⁡u1/2iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii-24u-4-4u-3)|)1/2)-1)du,
where γk=λk-1, k=1,2,….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 SpaceL2[0,1].Proposition 7.
Considerx→∞,
(46)P(∫01W^2(t)2dt>x)=(1+o(1))(2π)1/2(2z3/2,1)-2x-1/2exp⁡(-2z3/2,12x)·{720((-324z3/2,1-9+1328z3/2,1-7)sin2z3/2,1iiiiiiiiiiiiiiiiiiiii+(-325z3/2,1-10+926z3/2,1-8-127z3/2,1-6)iiiiiiiiiiiiiiiiiiiii×cos⁡2z3/2,1+325z3/2,1-10+326z3/2,1-8)}-1/2.Proof.
By Deheuvels [2] and Martynov [8], we have for all x>0(47)P(∫01W^2(t)2dt>x)=(1+o(1))(2π)1/2γ1-1(-D′(γ1))-1/2×x-1/2exp⁡(-γ1x2);
we take D(λ) and γ1=(2z3/2,1)2 into (47), and then the proof is completed.Proposition 8.
There exists a constantc>0 such that
(48)P(∫01W^2(t)2dt≤ε)=(c+o(1))ε-2exp⁡(-18ε),asε⟶0.Proof.
We start with proving (48) by recalling Li, 1992 [11, 12].
Given two sequencesak>0 and bk>0 with
(49)∑k≥1ak<∞,∑k≥1bk<∞,∑k≥1|1-akbk|<∞,
we have, as ε→0,
(50)P(∑k≥1akξk2≤ε)=(1+o(1))(∏k≥1bkak)1/2P(∑k≥1bkξk2≤ε).
By the asymptotic formula for zeros of Bessel function(51)z3/2,k=(k+12)π+O(k-1),k⟶∞,z5/2,k=(k+1)π+O(k-1),k⟶∞,
then ak=λk, b2k-1=((2k+1)π)-2, and b2k=((2k+2)π)-2, k∈N, which satisfy (49) and by the distribution identity ∫01W^22(t)dt=∑k≥1λ2k-1ηk2+∑k≥1λ2kη2k*2 and (50), there exists a constant c1, such that
(52)P(∫01W^2(t)2dt≤ε)=P(∑k≥1λ2k-1ηk2+∑k≥1λ2kηk*2≤ε)=(1+o(1))∏k≥1(bkak)1/2P(∑k≥1bkξk2≤ε)=(1+o(1))c1×P(∑k≥1ξ2k-12((2k+1)π)2+∑k≥1ξ2k2((2k+2)π)2≤ε)=(1+o(1))c1P(∑k≥1(k+2)-2ξk2≤επ2),asε⟶0.
Also, for alld>-1, there exists a constant c2>0, such that, as ε→0,
(53)P(∑k≥1(k+d)-2ξk2≤επ2)=(1+o(1))c2ε-dexp⁡(-18ε).
Connecting (52) with (53), we can obtain the proposition.



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