Extremes of L^{p}-norm of Vector-valued Gaussian processes with Trend

# Extremes of Lp-norm of Vector-valued Gaussian processes with Trend

Long Bai Long Bai, Department of Actuarial Science, University of Lausanne
UNIL-Dorigny, 1015 Lausanne, Switzerland

Abstract: Let be a Gaussian vector process and be a continuous function. The asymptotics of distribution of , the norm for Gaussian finite-dimensional vector, have been investigated in numerous literatures. In this contribution we are concerned with the exact tail asymptotics of with trend over . Both scenarios that is locally stationary and non-stationary are considered. Important examples include and chi-square processes with trend, i.e., . These results are of interest in applications in engineering, insurance and statistics, etc.

Keywords: Tail asymptotics; -norm; vector-valued Gaussian process; fractional Brownian motion; Pickands constant; Piterbarg constant.

AMS Classification: Primary 60G15; secondary 60G70

## 1. Introduction

In engineering sciences, extreme values of non-linear functions of multivariate Gaussian processes are of interest in dealing with the safety of structures, see [EXCRA1980] and the references therein. Probabilistic structural analysis to answer the question is: what is the probability that a certain mechanical (or other) structure will survive when it is subject to a random load. The load is then usually defined by some -dimensional vector process , and one seeks the probability that exceeds some more or less well-defined safe region, which is specific for the structure as

 (1) P{Y(t)∉Su(t),for some t∈[0,T]},

where the time-dependent safety region is defined by

 Su(t)={(x1,⋯,xn)∈Rn:∥x∥p≤h(t,u)}

with some continuous function and , the norm, i.e.,

 ||x||p={(∑ni=1|xi|p)1/p, p∈[1,∞),max(|x1|,…,|xn|), p=∞,

in the space .
Assume that where are independent copies of a centered Gaussian process which has continuous trajectories, variance function and correlation function and

 (2) d=(d1,…,dn),  1=d1=⋯=dm>dm+1≥dm+2≥⋯≥dn>0,1≤m≤n.

In the framework of (1), set , then we can rewrite (1) as

 P{∃t∈[0,T]Z(t)>h(t,u)}

where

 (3) Z(t):=Zp(t):=∥X(t)∗d∥p,

and hereafter, we call the norm process.
When , for a positive constant , as in the convention is called the chi process when and the chi-square process when .
Further, as the Gaussian processes, we can introduce the stationary, locally-stationary, and non-stationary norm processes according to the stationary, locally-stationary, and non-stationary properties of , respectively.
The investigate of

 P{∃t∈[0,T]Z2(t)>u}=P{supt∈[0,T]Z2(t)>u},as u→∞

is initiated by the studies of high excursions of envelope of a Gaussian process, see e.g., [BN1969] and generalized in [Lindgren1980a, Lindgren1980b, Lindgren1989]. When is stationary with and

 r(s,t)=1−a|t−s|α+o(|t−s|α),  |t−s|→0, α∈(0,2],

[Albin1990, Albin1992] develop the Berman’s approach in [Berman82] to obtain an asymptotic behavior of large deviation probabilities of the stationary chi-square processes.
Further, if there exists unique satisfies and

where and are positive constants related to , the tail asymptotic behavior of the non-stationary and are investigated in [Pitchi1994] and [FatalovA1993], respectively, under the application of the so-called ”double-sum method” in [Pit96].

Some recent contributions are focused on more general scenarios of chi process and chi-square process with , i.e.,

 P{∃t∈[0,T]Zc2(t)>h(t,u)}=P{supt∈[0,T](Zc2(t)+g(t))>u}, c=1,2,

where the continuous function is generally considered as a trend or a drift.
When are non-stationary Gaussian processes, , the non-stationary chi processes with trend, and , the non-stationary chi-square processes with trend, are studied in [EnkelejdJi2014Chi] and [PL2015], respectively.
When are locally-stationary Gaussian processes, [LJ2017] obtains the extreme of the supremum of with trend, see, e.g., [Berman92, Hus90] for more details about locally stationary Gaussian processes.

Considering both the locally stationary and non-stationary norm processes, the contribution of this paper concerns an exact asymptotic behavior of large deviation probabilities for with , constant and a continuous function, which contains the aforementioned results.
Organisation of the rest of the paper: In Section 2, the notation and some preliminaries are given. Our main results are displayed in Section 3. Following in Section 4 are two applications related to insurance and statistics. Finally, we present the proofs in Section 5 and several lemmas in Section 6.

## 2. Notation and preliminaries

First we introduce some notation, starting with the well-known Pickands constant defined by

where are constants and is a standard fractional Brownian motion (fBm) with Hurst index Further, define for non-negative continuous function.

 Pfα,a[−S1,S2]=E{supt∈[−S1,S2]e√2aBα(t)−a|t|α−f(t)},

and

 Pfα,a[0,∞)=limS2→∞Pfα,a[0,S2],  Pfα,a(−∞,∞)=limS1,S2→∞Pfα,a[−S1,S2].

The exact values of are known for and , namely,

 Pbt1,a[0,∞)=1+abandPbt22,a[0,∞)=12(1+√1+ab).

See [PicandsA, Pit72, debicki2002ruin, DI2005, DE2014, DiekerY, DEJ14, Pit20, Tabis, DM, SBK, GeneralPit16] for various properties of and .

Through this paper means asymptotic equivalence when the argument tends to or . We notice that denotes the tail distribution function of an random variable and .
For the norm process in (3) and a continuous function , we shall investigate the asymptotics of

 (4) P{supt∈[0,T](Zc(t)+g(t))>u},u→∞,

with a constant. As in [FatalovA1993, Pitchi1994], for , using the duality property of norm we find

 P{supt∈[0,T]Zc(t)>u}=P{supt∈[0,T]Z(t)>u1/c}=P{sup(t,v)∈[0,T]×SqY(t,v)>u1/c},

where is a centered Gaussian field defined on cylinder with

 (5) Sq={v∈Rn:||v||q=1},

where if , if and if .

###### Lemma 2.1.

On , attains its maximum at:
(i) for at points where ( stands at the i-th position), ( stands at the i-th position), ;
(ii) for at points on , ;
(iii) for at points , where

 z=(z1,…,zn), zi=±(di/d)2/(q−2),d=[n∑i=1d2p/(2−p)i](2−p)/2p,

( we take all possible combinations of signs ”+” and ”-” ), where .

The proof can be easily carried out by method of Lagrangian multipliers or referring to [FatalovA1993] [Lemma 3.1].
Next by [randomChaos2015], we have the following lemma.

###### Lemma 2.2.

For the norm process in (3), if for some , then we have that as

 P{Zc(t0)>u}∼Ψ(u1/cd)⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩2n(2−p)(1−n)/2, if  p∈[1,2),√2π2(2−m)2um−1cΓ(m/2)∏ni=m+1(1−d2i)−12, if  p=2,2m, if  p∈(2,∞],

with the convention and the same as in Lemma 2.1.

## 3. Extremes of Lp norm processes with trend

In this section, recall that in (3) is the norm process and ’s are independent copies of with continuous trajectories, variance functions and correlation functions .

### 3.1. Extremes of non-stationary Lp norm processes with trend

As in [Threshold2016], if is non-stationary, we introduce the following assumptions:

• attains its maximum on at the unique point and

 σ(t)=1−b|t−t0|β+o(|t−t0|β),    t→t0

for some positive constants .

• for some constants and

Further, we introduce a bounded measurable trend function which satisfies

• for some constants .

###### Theorem 3.1.

If assumptions (i)-(iii) are satisfied, then for in (2) and in Lemma 2.1, we have as

 P{supt∈[0,T](Zc(t)+g(t))>u}∼P{Zc(t0)>u}⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩u2α∗−2β∗a1/αd−2/αHα∫∞Qe−f(t)dt, if  α∗<β∗,Pf(t)α,ad−2[Q,∞), if  α∗=β∗,1, if  α∗>β∗,

where , , , and if , if .

###### Remarks 3.2.

i) In Theorem 3.1, if we assume that , we get the extremes of centered non-stationary norm processes i.e.,

 P{supt∈[0,T]Zc(t)>u}, u→∞.

ii) Following the similar arguments as in the proof of Theorem 3.1, the result in Theorem 3.1 still holds for if and if .

### 3.2. Extremes of locally stationary Lp norm processes with trend

If is locally stationary, as in [Threshold2016], we shall suppose that:

• where are positive continuous function on .

Before giving the scenarios with trend, we consider the extremes of the centered locally stationary norm processes.

###### Theorem 3.3.

Assume that , i.e., unit variance and covariance function satisfies assumptions (iv) and (v). Then we have for

 P{supt∈[0,T]Zc(t)>u}∼∫T0(a(t))1αdtd−2αHαu2αcP{Zc(0)>u}, u→∞,

where is the same as in Lemma 2.1.

###### Theorem 3.4.

Assume that , i.e., unit variance and correlation function satisfies assumptions (iv) and (v). Assume that is a continuous function which attains its maximum at a unique point satisfying assumption (iii) for some constants . Further, set and and is the same as in Lemma 2.1.
If , then we have as

where and if , if .
If , then we have

 P{supt∈[0,T](Zc(t)+g(t))>u}∼∫T0(a(t))1αeg(t)2d2dtd−2αHαu2α∗P{Zc(0)>u}, u→∞.

If , then we have

###### Remark 3.5.

By the proof, we notice that for the case in Theorem 3.4, the result always holds for any continuous function . When , the result holds for any bounded function .

###### Example 3.6.

For in (3) with the independent fractional Brownian motions, we have as

 P{supt∈[0,1]Z(t)−√1−t>u}=P{Z(1)>u}⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩u2α−2(12d2)1/αHα∫∞0e−f(t)dt, if  α<1,Pf(t)α,d−2/2[0,∞), if  α=1,1, if  α>1,

where and is the same as in Lemma 2.1

Following example is a special case of Theorem 3.4, which is corresponded with [LJ2017] [Theorem 2.1].

###### Example 3.7.

In Theorem 3.4, assume that , and is a continuous function, then we have

 P{supt∈[0,T](Z22(t)+g(t))>u}∼∫T0(a(t))1αeg(t)2dtHα21−m/2∏ni=m+1(1−d2i)−12Γ(m/2)um−12+1αe−u2, u→∞.

## 4. Applications

### 4.1. Ruin probability of a risk model

In theoretical insurance modelling a surplus process can be defined by

 U(t)=u+wt−X(t),   t≥0,

see [MR1458613], where is the initial reserve, is the rate of premium and the stochastic process denotes the aggregate claims process. See [rolski2009stochastic, DHJ13a, HX2007, ParisianBrownianfinite2017, Threshold2016, ParisianInfinite2018] for more studies on related risk models. Here we investigate

 X(t)=n∑i=1∣∣diBiα(t)∣∣2,t≥0,

where is the same as in (2) and are independent fractional Brownian motions. can be considered as the sum of independent claims or payments until time . The corresponding ruin probability over a finite-time horizon is defined as

 P{inft∈[0,1]U(t)<0}.

We present next approximation of this ruin probability.

###### Proposition 4.1.

We have as

 P{inft∈[0,1]U(t)<0}∼um/2−1e−u+w221−m/2Γ(m/2)n∏i=m+1(1−d2i)−1/2⎧⎪⎨⎪⎩u1/α−121−1/αHα, if  α<1,2, if  α=1,1, if  α>1.

Besides in risk modelling, the norm processes, especially the chi-square processes, are also widely utilized in hypothesis testing, see [HTDa1987, SCSP2014] and the reference there. Next we give an example.

### 4.2. The Ornstein-Uhlenbeck chi-square process in Quantitative Trait Locus detection

A Quantitative Trait Locus (QTL) denotes a gene with quantitative effect on a trait. The method used by most of geneticists in order to detect a QTL on a chromosome, is the Interval Mapping proposed by [LanderBO1989]. Using the Haldane distance and modelling in [Haldane1919], each chromosome is represented by a segment . The distance on is called the genetic distance. At each location , using the ”genome information” brought by genetic markers, a likelihood ratio test (LRT) is performed, testing the presence of a QTL at this position. [ADR2014] prove that when the number of genetic markers and the number of progenies tends to infinity, the limiting process of the LRT process is an Ornstein-Uhlenbeck chi-square process under the null hypothesis of the absence of QTL on the interval . In order to take decision about the presence of a QTL on , we need to calculate the critical value for the supremum of an Ornstein-Uhlenbeck chi-square process, i.e.,

 supt∈[0,T]S(t),

where the Ornstein-Uhlenbeck chi-square process is

 S(t)=n∑i=1Vi(t)2

and are independent identically stationary Gaussian processes with covariance function given by

 Cov(Vi(s),Vi(t))=e−2|t−s|.
###### Proposition 4.2.

We have as

 P{supt∈[0,T]S(t)>u}∼22−n/2Γ(n/2)Tun/2e−u/2.

## 5. Proofs

During the following proofs, are some positive constants which can be different from line by line and for interval we denote

 Lu(Δ1):=P{supt∈Δ1(Zc(t)+g(t))>u},  Lu(Δ1,Δ2):=P{supt∈Δ1(Zc(t)+g(t))>u,supt∈Δ2(Zc(t)+g(t))>u},

and

 Ku(Δ1):=P{supt∈Δ1Zc(t)>u},  Ku(Δ1,Δ2):=P{supt∈Δ1Zc(t)>u,supt∈Δ2Zc(t)>u}.

Proof of Theorem 3.1: We first present the proof for the case .
Set , with and for large enough

 Y(t,v)=n∑i=1diviXi(t),(t,v)∈R×Sq

with the same as in (5) which is a centered Gaussian field.
We have for some small and large enough

 (6) Lu([0,δ(u)])≤Lu([0,T])≤Lu([0,δ(u)])+Lu([δ(u),θ])+Lu([θ,T]).

We first give the upper bounds of and .
Set and . Then by Borell inequality as in [AdlerTaylor] and Lemma 2.2 for large

 (7) Lu([θ,T]) ≤ P{supt∈[θ,T]Z(t)>(u−gm)1/c} ≤ P{sup(t,v)∈[θ,T]×SqY(t,v)>(u−gm)1/c} ≤ exp⎛⎝−((u−gm)1/c−Q1)22V∗Y⎞⎠ = o(P{Zc(0)>u}), u→∞,

where and

 V∗Y:=sup(t,v)∈[θ,T]×Sq% Var(Y(t,v))≤(supt∈[θ,T]σ2(t))d2=σ2θd2

By assumptions (i) and (iii), we know that for some

 (8) u−g(t)σc(t)≥(u+w(1−ε1)|t|γ)(1+(1−ε1)bc|t|β)≥u(1+w(1−ε1)u|t|γ+(1−ε1)bc|t|β), (9) u−g(t)σc(t)≤(u+w(1+ε1)|t|γ)(1+(1+ε1)bc|t|β)≤u(1+w(1+ε1)u|t|γ+(1+ε1)bc|t|β)

hold for when small enough, then

 inft∈[δ(u),θ](u−g(t))2/cσ2(t) ≥inft∈[δ(u),θ]u2/c(1+w(1−ε1)u|t|γ+(1−ε1)bc|t|β)2/c (10) ≥u2/c+Q2(lnu)(ρβ)∨(ργ).

Denote with . By assumption (ii), we have that

 E⎧⎨⎩((n∑i=1divi¯¯¯¯¯Xi(t))−(n∑i=1div′i¯¯¯¯¯Xi(s)))2⎫⎬⎭ ≤2E⎧⎨⎩((n∑i=1divi¯¯¯¯¯Xi(t))−(n∑i=1divi¯¯¯¯¯Xi(s)))2⎫⎬⎭ +2E⎧⎨⎩((n∑i=1divi¯¯¯¯¯Xi(s))−(n∑i=1div′i¯¯¯¯¯Xi(s)))2⎫⎬⎭ ≤4E{n∑i=1(¯¯¯¯¯Xi(t)−¯¯¯¯¯Xi(s))2}+4E{n∑i=1(vi−v′i)2(¯¯¯¯¯Xi(s))2} ≤Q3|s−t|α+Q4n∑i=1|vi−v′i|2 ≤Q5(|s−t|α+n∑i=1|vi−v′i|α)

holds for and . Thus it follows from [Pit96] [Theorem 8.1], (5) and Lemma 2.2 that

 Lu([δ(u),θ]) ≤P{sup(t,v)∈[δ(u),θ]×Sqn∑i=1divi¯¯¯¯¯Xi(t)>infs∈[δ(u),θ](u−g(s))1/cσ(s)} ≤Q6u2(n+1)αΨ(infs∈[δ(u),θ](u−g(s))1/cdσ(s)) ≤Q6d√2πu2(n+1)α−2cexp(−12d(u2/c+Q2(lnu)(ρβ)∨(ργ))) (11) =o(P{Zc(0)>u}), u→∞.

Thus by (7), (5) and the fact that for positive, we have

 (12) Lu([δ(u),θ])=o(Lu([0,δ(u)])),  Lu([θ,T])=o(Lu([0,δ(u)])), u→∞,

which combined with (6) imply

 (13) Lu([0,T])∼Lu([0,δ(u)]), u→∞.

Now we focus on the asymptotic of , as .
Denote for any and some

 Ik(u)=[ku−2/α∗λ,(k+1)u−2/α∗λ],  k∈N,  N(u)=⌊(lnu)2qβ∗u2α∗−