Asymptotic Distribution Of Sum And Maximum For Strongly
Dependent Gaussian Processes
William P. McCormick
University of Georgia, Statistics Department
Athens, GA 30606 USA
bill@stat.uga.edu
Hwai-Chung Ho
Academia Sinica, Institute of Statistical Sciences
Taipei, Taiwan ROC
hcho@stat.sinica.edu.tw
The problem of determining the asymptotic distribution of (Sn, Mn), where Sn = ' Xj and Mn =
w Xj with w denoting maximization, has been of considerable interest and the subject of several papers.
We mention Hsing (1995) as a recent reference and refer to that paper for further results on the topic.
The main result in Hsing (1995) is that for strong mixing sequences, such that Sn satisfies the central
limit theorem, asymptotic independence of (Sn, Mn) ensues.
Gaussian sequences have long been studied with regard to the asymptotic properties of extreme
values. It is well known that for stationary Gaussian sequences 6 Xn > with E Xn = 0 and E Xn2 = 1 that
r(n) = E X0 Xn = o(1/ln n) is the critical rate of decay of the correlation function to ensure that
asymptotically the distribution of Mn behaves exactly like that of Mn* = w Xj*, where 6 Xn* > is an i.i.d.
sequence of standard normal r.v.’s. When r(n) ln n 6 4 and certain regularity conditions hold,
behaviour different from the i.i.d. case appears for the maximum. See Leadbetter et.al. (1983) for a
discussion of these issues.
Ho and Hsing (1996) considered the asymptotic distribution of (Sn, Mn) for stationary Gaussian
sequences. They established asymptotic independence when r(n) = o(1/ln n) and a regularity condition
holds for r(n). This is interesting because, in general, stationary Gaussian sequences with correlation
function r(n) satisfying r(n) = o(1/ln n) will not be strong mixing. Thus, Gaussian sequences provide
a natural class of non-strong mixing sequences to study for the asymptotic behaviour of (Sn, Mn) and,
as the Ho and Hsing (1996) result shows, strong mixing is not essential for asymptotic independence.
When r(n) ln n 6
0 ø 0, 4‚, Ho and Hsing (1996) show that (Sn, Mn) are asymptotically dependent and
obtain the asymptotic joint distribution. This result provides a useful class of models for the joint
distribution of (Sn, Mn).
By the devise of first centering the maximum by subtracting the mean, we show that asymptotic
independence of (Sn, Mn - X̄n) follows both under the case of r(n) ln n = o(1) and ( r(n) ln n )-1 = O(1)
and certain regularity conditions hold on r(n). Continuous time results are also obtained for the joint
distribution of ( ST, MT ) where ST = *0T Xt dt and MT = w0#t#T Xt for a class of continuous time Gaussian
processes.
n
For the class of Gaussian processes of the form Xt =
∑
i
i =1
(t)Zi, t $0 where Z1, ..., Zn are i.i.d.
n
∑
N(0,1) r.v.’s and
i =1
2
i
(t) /1, we obtain the following approximation to the joint distribution of (ST,
MT ). Let ci = *0T i(t)dt and di = ci /%('1n ci2 ), 1 # i # n. Define Ju = 6 t 0 [0,T] : Š d, (t) Ž $u >, where
d = (d1, ...,dn ) and
+ 1/%(2
)y e!1/2y2
(t) = (
(t), ...,
1
n
(t) ). Then P{ ST/ %Var(ST) $x, MT $y} µ 1/2 length ( Ju )e!1/2y2
as x, y 6 4 with x/y 6u 0 (0,1].
REFERENCES
Ho, H. C. and Hsing, T. (1996). On the asymptotic joint distribution of the sum and maximum of
stationary normal random variables. J. Appl. Prob. 33, 138-145.
Hsing, T. (1995). A note on the asymptotic independence of the sum and maximum of strongly mixing
stationary random variables. Ann. Prob. 23, 938-947.
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983). Extremes and Related Properties of Stationary
Sequences and Processes. Springer-Verlag, N.Y.
RESUME
Soit, 6 Xn > une suite Gaussienne stationnaire de variables aleatoires normales centre3 es re3 duites avec
n
une fonction de covariance r(n) = E X0Xn .
Soit X̄n = (1/n)
∑
i =1
Xj . Sous certaines conditions
modere3 es de regularite3 sur r(n) et sous la condition que r(n) = o(1) ou ( r(n) ln n )-1 = O(1), la
distribution asymptotique de (Sn, Mn - X̄n ) with Sn = ' Xj and Mn = w Xj est obtenue. Aussi, les
re3 sultats en temps continu ainsi qu’une formule d’approximation de la surface de la queue a la loi du
couple forme3 par la somme et le maximum sont presente3 s.