Extreme values of a Gaussian process

EXTREME VALUES OF A GAUSSIAN PROCESS Y.

K.

CHOI,

K.

S.

HWANG AND

S. B.

KANG

1. Introduction and Results

Let {X (t) : 0

~

t < oo} be an almost surely continuous Gauss- ian process with X(O) =

^0,

E{X(t)} =

⁰

and stationary increments E{X(t) -X(s)}2

=

u

²

(lt-sl), where u(y) is a function of y

~ 0

(e.g., if {X (t);

0~

t < oo} is a standard Wiener process, then u(

t)

= Vi). As- swne that u(t), t > 0, is a nondecreasing continuous, regularly varying function at infinity with exponent

'Y

for some

0

<

^'Y

<

^1.

A positive function u( t), t > 0, is said to be regularly varying at infinity with

exponent

'Y

> 0 if, for all x > 0, one has

lim

u( xt) = x'Y.

t-oo

u(t)

Let

^aT

(0 < T <

⁰⁰⁾

be a function of T for which (i)

aT

is nondecreasing,

(ii) 0 <

^aT ~

T,

(iii) T /

^aT

is nondecreasing, and denote, for e < T <

00,

and

( 2 )-1/2

f3T = 2u (aT)(log(T/aT) + loglogT) .

Received September ·27, 1994.

1991 AMS Subject Classification: 60F05, 60G05.

Key words: Wiener process, Gaussian process, regularly varying function and Borel-Cantelli lemma.

This work was supported by the Foundational Juridical Person of Gyeongsang National University Research and Scholarship Foundation, 1993.

740 Y.K. Choi, K.S. Hwang and S. B. Kang

Define continuous parameter processes G1(T), G

₂

(T), ... , G1o(T) by G1(T) = ^{sup sup IX(t} + s) - X(t)l,

0:5s:5aT O:5t:5T-s

G

2

(T)= sup sup (X(t+s)-X(t»,

O:5s:5aT O:5t:5T - s .

G

3

(T) = sup sup IX(t + ^{s) - X(t)l,}

0:5s5aT 0:5t:5T - aT

G

4

(T) = ^{sup sup (X(t} + s) - X(t»,

0:5s:5aT 0:5t:5T-aT

Gs(T) =

^SUp

^IX(t +

^{aT) -}

^X(t)l,

0:::;t:5T-aT

G

6

(T)

=

sup (X(t +

^{aT) -}

^X(t»,

. 0:5t:5T-aT

G 7 ^(T}

^=..SU~.

^IX(T

^-t~S}

^=-X(T)/,.

0:5s:5aT

Gs(T) = ^{sup (X(T} + S) - X(T»,

O:5s:5 aT

Gg(T) = ^IX(T +

^{aT) -}

^X(T)I,

Glo(T) = X(T +

^{aT) -}

^X(T),

respectively. Clearly, G1(T) is the largest process and G1o(T) is the smallest one of all Gi(T), i = 1,2,· .. ,10.

Our main object of this paper is to obtain the almost sure limiting values of Gi(T), i

=

1,2,· .. ,10, under the varying conditions on

^aT.

Thus we are concerned only with behaviors of functions near at infinity.

In our proof we shall

use

the small letter c for a positive constant which may be different from line to line if necessary.

An upper bound for OtTGi(T) is estimated

as

follows:

THEOREM

1.1. Let {X(t) : 0 $ t < oo} be an almost surely con- tinuous Gaussian process with X(O) = ^O,E{X(t)} = ^{0 and E{X(t)-}

. X(s)}2 = ^u

²

^{(\t - sI). Assume that u(t),t > 0, is a non decreasing con-}

tinuous, regularly varying function at

⁰⁰

with exponent "'( for some

o < "'( < 1. Let

aT

(0 <T < (0) satisfy the conditions such that

(i)

^aT

is nondecreasing,

(n) 0 <

^aT

$ T,

(iii) T/aT is nondecreasing and

(iv) limT_oc (logT -logaT)/loglogT = r, 0

~

r

~ ^00.

Tben webave

limsup

_T-oc aT

Gj(T) ~ ^Jl _+r

^r

^a.s.

wbere i = 1,2, ... ,10.

When

r

<

^00,

this theorem does not hold if we substitute PT for

QT,

and thus the QTis a critical normalizing factor to get the theorem.

It is interesting to compare this theorem with the "limsup" theorems of Cs8.ki et al. [2) and Choi [1].

The next theorem is easily proved by the same way as the proof of Theorem 1.1:

THEOREM

1.2. Let X(t) and O'(t) be as in Tbeorem

^1.1.

Let aT (0 < T < (0) satisfy the conditions (i), (ii) and (iii) of Theorem

1.1

and tbe condition

(iv)' limT_oo (log T -log aT )/log T =

^r,

0

~ r _~

1.

Then we have

limsup

aT

Gj(T) ~ ^J1

^{r 8.S.}

T-oc

+

r

where i = 1,2, ... ,10.

Sufficient conditions for lower bounds of PTGj(T) concerning "lim- inf' are obtained:

THEOREM

1.3. Let {X(t) : 0

~

t < oo} be an almost surely con- tinuous Gaussian process with X(O) = ^{0, E{X(t)}} = ⁰ and E{X(t) - X(s)}2 = 0'2(lt - sI). Assume tbat O'(t),t > 0, is a nondecreasing con- tinuous, regularly varying function at

00

with exponent

'Y for

some

o <

'Y

< 1. Let aT (0 < T < (0) satisfy the conditions such tbat (i) aT is nondecreasing,

(ii) 0 < aT

~

T,

(iii) T/aT is nondecreasing

and

742 Y. K.Choi, K. S. Hwang and S. B. Kang

(iv)

limT->oo

(logT -logaT)/loglogT = ^{r, 0 :S r :S}

^00.

Assume that for any a :S b :S c :S d

(v) E{ (X(b) - X(a)) (X(d) - X(c))} :S 0, (or, u 2(t) is co cave for

t > 0). .-

Then we have

liminf

_T->oo

PT Gi(T) ~ J _{1 +} ^r

_r

^a.s.

where i = 1,2, ... ,6 if

r

> 0; and i = 1,3,5, 7,9

ifr

= O.

THEOREM

1.4. Let X(t) and u(t) be as in Thoerem 1.3. Let aT (0 < T < (0) satisfy the conditions such that

(i) aT is

nondecreasing~

(ii) 0 < aT :S T,

(iii) T / aT -is non decreasing and

(iv)"

limT->oo

(logT -logaT)/loglogT

= r,

1 <

r

:S

00.

Assume that for t > 0,

(v)' (T2(t), t > 0, is twice continuously differentiable which satis:fies

Then we have

liminf

_T->oo

PT Gi(T) ~ J1 _+r ^{r a.s.}

where i

=

1,2, ... ,6

For instance, we can choose aT as 1, logT, TB (0 < () < 1),

T/(logT)T (0 < r < (0) and cT(O < c:S 1), etc. These theorems show us that we can get exact lower bounds of PTGi(T) according to the functions aT to be chosen. Of course, if we replace PT in Theorems 1.3 and 1.4 by

aT,

then they do not hold.

By using the proof process of Theorems 1.3 and 1.4, one can easily

obtain the following

THEOREM 1.5. Let X(t) and O'(t) be as in Thoerem 1.3. Let aT (0 < T < 00) satisfy the conditions such that

(i) aT is non decreasing, (ii) 0 < aT

~

T,

(iii) T / aT is non decreasing and

(iv)' limT-+oo (log T -log aT )/log T =

r,

0

~ r ~ 1.

Assume that either, for any

a _~

b

~ c _~

d

(v) E{ (X(b) - X(a)) (X(d) - X(c))}

~

0, ( or, u

²

(t) is concave for

t

> 0)

or

(v)' 0'2(t),

^t

> 0, is twice continuously differentiable which satisfies

Then we have

liminf 0T

_T-+oo Gi(T)

2: J _{1 +}

^r _r

^a.s.

where

i =

1,2,··· ,6 if 0 <

r ~

1; and

i

= 1,3,5,7,9 if

r

= O.

Combining Thoerems 1.2 and 1.5, we have a "limit" value:

THEOREM 1.6. Let

X(

t), u( t) and aT be as in Thoerem 1.5. Further assume that the conditions (v) and (v)' of Theorem 1.5 are satisfied.

Then we have

lim

0T

Gi(T) =

J

^r

^a.s.

T ... oo 1

+

r

where i

=

1,2,,··,6 HO < r

~

1; and i

=

1,3,5,7,9 Hr

=

O.

If O'(t)

=

fY(O < 'Y < 1), then the process {X(t);O

~

t < oo} is a fractional Brownian motion of index 'Y. It is clear that in case of

o ^<

^'Y ~

1/2, the condition (v) of Theorem 1.5 is satisfied, and if

o < 'Y < 1 then the condition (v)' of Theorem 1.5 is satisfied. Hence,

Theorem 1.6 can be applied to every fractional Brownian motion.

744 Y. K.Choi, K. S. Hwang and S. B. Kang

2. Proofs

When r

= 00

in our theorems, let us define r j (1 + r) by 1, then the results immediately follow from Csaki et al. [2] and Choi [1]. IT

r =

0 in Theorems 1.3, 1.5 and 1.6, the results are obvious. So we shall prove the theorems when 0 <

r

<

^00.

The following lemma is essential to prove Theorem 1.1:

LEMMA 2.1. (Choi [11) LetX(t) andu(t) bea.sin Theorem 1.1. Let

aT

(0 < T <

⁰⁰⁾

satisfy the conditions (i), (ii) and (iii) of Theorem

1.1.

Set

G( ) _ X(t + ^{s) -} X(t) . s, t -

_{u aT}

( ) ,

Then for any small

^E

> 0 there exist constants To

=

To(

^E)

^and

^CE

depending only on

^E

such that for all u 2:: 0 and T 2:: To

P {sup sup G(s,t) 2:: u} ~ ^c

^E

(:!.-)e-

^u2/(2+E).

O$s$aT O$t$T-s aT

Proof of Theorem 1.1. Set 8 = ^(r + E)j(1 + r) for any small

E

> O.

Applying Lemma 2.1, we have, from the condition (iv), P {

OtT

G1(T) > y'8(1 + E)}

< 2P{ " X(t+s) -XCi)

_ sup sup

O$s$aT o$t$T-s u(aT)

> y'20(1 + E){log(TjaT) + ^10gT}}

~ CE(~) exp{- ^{2: /28(1 +} EHlog(TjaT) + 10gT})}

= _{CE(~) (~;)}

-29(HE)/(2+E)

(

T

)1-{(2+2E)/(2+E)}{(r+E)/(1+r)}

=

C

E _"_ T-9(2+.2E~/(2+E)

aT "

~ CE

(log T)

(r+E){1-{(2+2E)!(2+E) }{(r+E)/(Hr)}}T-8(2+2E)/(2+ E)

~ ^C_E

(log T)

⁹

T-

8(1 +{E/ (2+E)})

~ c_E

T-

8/ 2

provided T is big enough. For given kEN, let TA:

=

exp( k

^Q)

where 1/2 < a <

1.

Then the above statement gives

The series

LP{aT.G1(Tk) > \1"0(1 +e)}

k

is convergent, and using

th~

Borel-Cantelli lemma, we have

The remainder of the proof is to show that

(2.1) limsup aT G1(T) :5limsup aT. G1(Tk)'

T-oo k-oo

Let T be in Tk-l :5 T :5 Tk. Then, by the conditions (i)

^{I V}

(iii),

(2.2)

From the conditions (i)

^{I V}

(iii) and using the fact that (log u) / u is decreasing for u > e, we have

and

746 Y.K. Choi, K. S. Hwang and S. B. Rang

Thus it follows from the regularity of u(.) at

00

that 1 > ^U(aTk_l) > ^u(exp{-a(k -1)'~-l}aTk)

- u(aTk) - u(aTk)

(2.4)

~

exp{ -a(k - I)Q-l(-y + ^7J)}

^-+

1 as k

-+ 00,

where 7J > 0 is small enough. Combining (2.2), (2.3) and (2.4), we obtain the inequality (2.1). This completes the proof of Theorem 1.1.

For proving Theorem 1.3, we need the following lemmas:

LEMMA

2.2. (Slepian [4]) Let G(t) and G*(t), 0

~

t <

^00,

be cen- tered Gaussian processes, possessing continuous sample path functions, with E{G(t)}2 = ^E{G*(t)}2 = ^{1, and} let p(s,t) and p*(s,t) be their respective covariance functions. Suppose that we have

p(s,t)

~

p*(s,t), o

~

s, t <

^00.

Then p{ ^{sup G(t)} ~ u} ~ p{ ^{sup G*(t)} ~ u}.

09:5T 0:5

^t

:5T

LEMMA

2.3. (Choi [1]) Let X(t), u(t) and aT be as in Theorem 1.3. For 0 < a < 1 set Tk

=

exp(k

^{Q) ,}

kEN, and let T be

in

T

k ~

T

~

Tk+I. Then we have

Proof of Theorem 1.9. For given T large, let us define a positive integer hT by hT = [TjaT], where [x] denotes the greatest integer not exceeding x. It is clear from the conditions (iv) that h

_T

is increasing.

For i = 1,2, ... ,hT, we define the incremental random variable ZT(i) = X(iar) - X«i -l)aT).

Clearly ZT(i)ju(aT) is a standard normal random variable. By the condition (v), we have

covariance(ZT(i), ZT(j»

~

0, i f= j.

For any small

E

> 0, we set

r -

2€

b= > O.

(1 - (€/2))(1 +

r -

E)

Applying Lemma 2.2 for G*(i)

⁼

ZT(i)/a(aT), i

⁼

^1,2,··· ,hT, we have

P{PT G

6(T)

< Jb(l - €)}

~ p{ ^{sup Z(T(j)) <} J2b(1 - €)(log(T/aT) + loglogT)}

l~j~hT

a aT

~ {<P(UT)}

hT

where UT

=

J2b(1 - €)(log(T/aT) + log log T) and <p(.) denotes the standard normal distribution function. Since, for large T,

and for some c > 0

1

2

P(Z _~

UT)

~

cexp( -2"UT ) ( T 1 T)

-b(l-E)

=

C -

og ,

aT we have

{<P(UT)} h

^T

~ ^{exp {} -c(~) (~ ^log T)

^-b(I-E)}

{ (

T ) I-b(I-(E/2» }

~

exp -c aT (log

T)-b(I-(f/2») .

Using the condition (iv), we get

- T

~

(log T)

r-E

aT and

{<p( UT) }

hT ~

exp{ -c(log T)(

r-f){1-b(I-(E/2» }-b(l-(£/2»)}

=

exp{ -c(log

T)E}.

748 Y. K. Choi, K. S. Hwang and S. B. Kang

Therefore we obtain, for all large T,

p{I1TG

6

(T) < y'b(l- e)} _{~ exp{} -c(logTt}.

For 0 < a < 1, set Tk

=

exp(kO!),k

E

N. Then

P{I1TA;G

₆

(Tk) < y'b(l- e)}

~

exp( _CkO!E) and the series

LP{I1TA;G

6

(Tk) < y'b(l- e)}

k

is convergent. The Borel-Cantelli lemma implies liminf

_k-oo

f3TA;G

6

(Tk) ~ VI ₊

^r _r

^a.s.

Let T be in Tk

~

T

~

Tk+1 for given Tk. Then Lemmma 2.3 completes the proof of Theorem 1.3.

The next lemmas are applied to prove.Theorem 1.4:

LEMMA 2.4. (Leadbetter et al. [3]) Let {Xi; i = 1,2"" ,n} be jointly standardized normal random variables with covariance (Xi, X

j )

=Aij

such that

b = max

I Ai" I < 1.

i:f=j

Then for any real number u and integers 1

~ 1₁

<

12

< ... < lk

~

n with k

~

n,

P{l$"J$k X'j ~ ^u} ~ ^{'P(u)}k

(2.5) + ^K 1~~9Ir ^{.;1_(- 1 +U:}

^ri)

where

^rij

=

Al;lj

and K = K(b) is a positive constant depending on

b

but not n, u and k, and 'P(.) denotes the standard normal distribution function.

We shall estimate an upper bound of the second term in the above inequality (2.5) by imposing a stationary condition concerning covari- ance functions of {Xi; i = 1,2"" ,n}. Note that in the condition (iv)"

of Theorem 1.4, when T > 0 is a large number, one can choose a big integer M > 0 such that M < (log T)B < T /

^aT

for some B > O.

The following lemma is easily verified by the same way as the proof of

Lemma 4.4(ii) in Choi [lJ:

LEMMA

2.5. Let Xi(i

=

1,2"" , n), 8 and rij be given as in Lemma 2.4. Assume that the covariance functions rij be such that

Irij

¹_~

Pli-jl < 1 for i

=1=

j.

Suppose that the function aTe 0 < T <

⁰⁰⁾

is as in Theorem 1.4. For given T > 0 large, set k

=

[T j (M aT ) J, where [x] denotes the greatest integer not exceeding x, and let, for some v > ^0,

Pm<m-

^II

for all m=li-jl=I,2,···,k-1.

For 0 < r <

00,

let u

=

J2b(1- e){log(TjaT) + loglogT} and b

=

(r - 2e)/{(1 - (e/2))(1 + r - en ^>

⁰

^{for any small}

^€

^{> O. Then there}

exists K > 0 depending only on e,8 and 1/ such that

where 80

= {{

rl/(l +r )(1-8)} j {(I + 1/)( 1+8)( 1- (ej2))(1 +r - _{e)}} - e'} and e' >

0

is 3mall enough.

Proof of Theorem 1.4. Let 1 < r <

00

in the condition (iv)".

Then, for T > 0 large, we can choose a big integer M > 0 such that M < (logT)B < TjaT for some B > O. Define a positive integer k T by k _T = [Tj(MaT)] as in Lemma 2.5. By (iv)", k _T is increasing. For i

=

1,2" ..

,kT,

we define the incremental random variable

YT(i) = X(MiaT) - X((Mi - l)aT).

Then YT(i)jO'(aT) is a standard normal random variable. It follows that for large T > 0,

(2.6)

P{I3

T

G

6

(T) < Jb(l - e)}

~p{ ^{sup Y(T(i))} <J2b(1-e){lOg(T j a

T

)+loglogT}}.

l~i~kT 0'

aT

Let rT(i,j) = correlation(YT(i), YT(j)), i

⁼¹⁼

j, and let m = li - jl

~

1.

By the same process as the proof of Theorem 2.2 in Choi [1] (Here we

750 Y. K. _Choi~K.S. Hwang and S. B. Kang

make use ofthe condition (v),), we can get IrT(i,j)1 < m-v where v =

1-

'Y

> O. Applying Lemmas 2.4 and 2.5 for X'j

=

YTU)/u(aT),

j =

1,2"" ,kT, and UT = .J2b(1- e){log(T/aT) +loglogT}, the last term of (2.6) is less than or equal to

{~(UT)}kT

+ K(logT)-6

^o

where li o = {{rv(1+r)(1-li)}/((1+v)(1+li)(1-(€/2»(1+r-€)}}-€' and e' > 0 is small enough. Thus we have

P{.8TG

6

(T) < .Jb(l - e)} ~ exp{-c(log TY} + K(log T)-6

^0•

This inequality is bounded by K(logT)-6

o •

For given kEN, let us set Tk = exp(k

^{Q) ,}

where a is taken by

_ (1 + v)(l + li)(l- (£/2»(1 + r - e) , 0 1 > a - _rv ( ) (

₁

+

^{r 1-}

u 1:) +. e > .

Then we have

and the series

L ^P{.8T

^k

^G

⁶

^{(Tk) < .Jb(l- €)}}

k

is convergent, and hence the Borel-Cantelli lemma implies

Letting T be in Tk ::; T

^:$

Tk+l for given T k, Theorem 1.4 immediately follows from Lemma 2.3.

References

1.Y. K. Choi, Erdiis-Renyi type laws applied to Gaussian processes, J. Math.

Kyoto Univ., 31-1 (1991), 191-217.

2. E. Csaki, M. Csoro, Z. Y. Lin and P. Revesz, On infinite series of indepen- dent Ornstein-Uhlenbeck processes,Stoc. Processes and their Appl., 39(1991), 25-44.

3. M. R. Leadbetter, G. Lindgrem and H. Rootzen, Eztreme, and related proper- tie, of random ,equence, and procelles, Springer-Verlag, 1983.

4. D. Slepian, The one-,ided barrier problem for Gaw,ian noise, Bell. System Tech. J., 41 (1962), 463-501.

Department of ma.thematics College of NaturaJ Science Gyeongsang National University Chinju 660-701, Korea

E-mail:[email protected]