Csorgo and Revesz (1979) proved a weaker version, in terms of an upper bound for the probability but not its precise asymptotics, of (22) on the increments of Brownian motion. Noting that X(t) := (t2 — t\)~ll2{B{t2) - B{t\)] is a Gaussian random field on the domain {£ = (£i,£2) : 0 < £1 < £2}:
Chan and Lai (2003c) considered more general Gaussian random fields.
Moreover, in order to be able to extend the result from the Gaussian ran- dom field to some process close to it in a certain sense (such as moderate deviations), they came up with the following formulation of "asymptotically Gaussian" random fields and developed asymptotic formulas for related boundary crossing probabilities, of which (22) is a special case. For vectors t,u € Rd, the relation t < u denotes U < m for all i and t < u denotes U < Ui for all i. For C > 0, let Jt,< = I\Li [*ằ. U + 0 - L e t W ( u ) : u € Rd} be a continuous Gaussian random field such that
Wt(0) = 0, E[Wt{u)} = - | |u| r rt( u / | M | ) / 2 and cov{Wt{u),Wt{v)) = [||u||art(tt/||ti||) + ||i;|rrt(t;/||t;||)
-\\u-v\\art((u-v)/\\u-v\\)}/2, (23) where || • || denotes Euclidean norm and rt : Sd _ 1 —> R+ is a continuous
function satisfying
sup \rt(v) — ru(v)\ —• 0 a s u - > t . (24) For c > 0, let Xc be random fields such that EXc{t) = 0, EX%{t) = 1
for all c and £. Let D be such that [D]s := {£ + u : t S D, \\u\\ < 6} is a subset of the domain of Xc for some 5 > 0 and all c large enough. Define pc(t,u) = E[Xc(t)Xc(u)\ and assume that
(C) pc(t, t + u) = 1 - (1 + 0(l))HaL(|M|)rt(ô/|M|)
uniformly over £ e [D]$ and compact sets of u / Ac > 0, and that the following conditions also hold uniformly over £ G [D]s for c large enough:
(Al) P{Xc(t) > c - y/c} ~ ^(c - y/c)
uniformly over positive, bounded values of y. The convergence in (24) is assumed to be uniform in £ € [D]g, with supte[£)]J)1,es'i-1 rt(v) < °°> a n (i for a > 0 and positive integers m,
{c[Xc(t + akAc) - Xc(t)} :0<ki< m}\Xc{t) = c-y/c
^ =4> {Wt(ak) :0<ki<m}
uniformly over positive, bounded values of y, where we use "\Xc(t) = c — y/c" to denote that the distribution is conditional on Xc(t) = c — y/c. In addition, there exists a positive function h such that limy_*oo h(y) = 0 and (A3) P{Xc{t + uAc) > c - 7/c, Xc(t) < c - y/c} < h{y)ijj{c)
for all u > 0 and 7 > 0, and there exist non-increasing functions Na on R+
and positive constants ja such that 7a —> 0 and Na(ja) + Jx wsJVa(7a + u) duj — o(ad) as a -* 0, and
(A4) P{ sup Xc(t + uAc) > c, Xc(t) < c - 7/c} < iV0(7)V(c),
0 < u < a
for all 7a < 7 < c and s > 0. Moreover, there exists a non-increasing function / : [0,oo) -> R+ such that /(||r||) = 0(e_llrHP) for some p > 0 and for all 7 > 0 and c sufficiently large,
(A5) P{Xc(t) > c - 7/c, Xc(t + uAc) > c - 7/c} < rl>(c - 7/c)/(||u||) uniformly in t and t + uAc belonging to [D]$.
Note that (Al) says that the "moderate deviation" event {Xc(t) >
c — y/c} has probability like that of a standard normal. Whereas this refers to the marginal distribution of Xc(t), the joint distribution is assumed in (A2) to be asymptotically normal in the sense of weak convergence for local increments conditioned on Xc(t) = c—y/c. Note that the same a, L(-) and rt(-) appear in (C) and the mean and covariance functions (23) of the limiting Gaussian field Wt(-) in (A2). In fact, if Xc = X is a Gaussian field satisfying condition (C), then (A2) holds; see Corollary 2.7 of Chan and Lai (2003c). Assumptions (A3)-(A5) are mild technical conditions under which the probability of s u pu € / t KA Xc(u) exceeding c can be computed via (Al) and (A2), yielding the following asymptotic formulas.
Theorem 5.1. (i) Let K > 0. Assume (C) and (Al)-(A4). Then
P{ sup Xe(u) > c} ~ V(c)[l + HK(t)] (25)
ô 6 f l , K 4c
uniformly overt G [D}$, where HK(t) = /0°° e3'P{sup0<u.<Ar Wt(u) > y} dy is finite and uniformly continuous int £ [D}$.
(ii) Assume (C) and (Al)-(A5). Then H{t) = limK-_f0Oii'-dHK-(t) exists and is uniformly continuous and bounded below on D. Moreover, as c —ằ 00 and Êc —> 00 such that lc = o(A~1),
P{ sup Xc{u) > c} ~ edci(>(c)H(t), (26)
ueit,tcAc
P{ sup Xc{u) > c, sup Xc(v) >c} = o(^V(c)), (27)
uElt,ecAc u € B \ / t , <c A c
uniformly over t £ D and over subsets B of [D]g with bounded volume.
Dividing (26) by (£cAc)d, which is the volume of It,ecAc, yields an asymptotic boundary crossing "density" A~dip(c)H(t) of Xc at t. By in- tegrating this "density" over D, or more precisely, by summing (26) over the "tiles" It,ecAa of D and applying (27) together with the fact that D is bounded and Jordan measurable, it follows that
P{snpXc(t) > c} ~ ^(c)A~d [ H(t) dt. (28)
te.D JD
Chan and Lai (2003c) also extend these results to the case when the maxima are over sets Dc that grow with c and to more general boundary crossing probabilities. The normalized moving averages (*2 —
£i)~1//2[W(t2) - W(ii)] in (22) are also simple prototypes for signal de- tection problems, in which the Brownian motion W(t) is replaced by a Gaussian field X(t) and £2 — t\ is replaced by var(X(t2) — -X^ti)) with multidimensional index t; see Bickel and Rosenblatt (1972), Siegmund and Worsley (1995) and Adler (2000). Again conditions (C) and (Al)-(A5) can be shown to hold for these applications and also for their discrete-time analogues (like 5[ntj in place of W(t) in (22); see Chan and Lai (2003c)).
References
1. R. J. Adler. On excursion sets, tube formulas and maxima of random fields.
Ann. Appl. Probab. 10, 1-74 (2000).
2. M. Basseville and I. V. Nikiforov. Detection of Abrupt Changes: Theory and Applications. Prentice-Hall, Englewood Cliffs, (1993).
3. A. Benveniste, M. Basseville and G. Moustakides. The asymptotic local ap- proach to change detection and model validation. IEEE Trans. Automatic
Control 32, 583-592 (1987).
4. P. J. Bickel and M. Rosenblatt. Two dimensional random fields. In Multi- variate Analysis III (P.R. Krishnaiah, ed.), 3-13. Academic Press, New York.
(1973).
5. N. H. Bingham and C. M. Goldie. Riesz means and self-neglecting functions.
Math Z. 199, 443-454 (1988).
6. N. H. Bingham and L. C. G. Rogers. Summability methods and almost- sure convergence. In Almost Everywhere Convergence II (A. Bellow and R.L.
Jones, eds.), 69-83. Academic Press, New York, (1991).
7. H. P. Chan and T. L. Lai. Saddlepoint approximations and nonlinear bound- ary crossing probabilities of Markov random walks. Ann. Appl. Probab. 13, 394-428 (2003a).
8. H. P. Chan and T. L. Lai. Moderate deviation approximations for the in- crements of Markov random walks and their applications to detection of structural changes. Tech. Report, Dept. Statistics, Stanford Univ. (2003b).
9. H. P. Chan and T. L. Lai. Maxima of Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of ran- dom variables with multidimensional indices. Tech. Report, Dept. Statistics, Stanford Univ. (2003c).
10. Y. S. Chow and T. L. Lai. Limiting behavior of weighted sums of independent random variables. Ann. Probab. 1 810-824 (1973).
11. Y. S. Chow and H. Teicher. Probability Theory, 2nd ed. Springer-Verlag, New York, (1988).
12. M. Csorgo and P. Revesz. How big are the increments of a Wiener process?
Ann. Probab. 7, 731-737 (1979).
13. M. Csorgo and P. Revesz. Strong Approximations in Probability and Statis- tics. Academic Press, New York, (1981).
14. A. de Acosta and J. Kuelbs. Limit theorems for moving averages. Z.
Wahrschein. 64, 67-123 (1983).
15. P. Deheuvels, L. Devroye and J. Lynch. Exact convergence rate in the limit theorem of Erdos-Renyi and Shepp. Ann. Probab. 14, 209-223 (1986).
16. P. Erdos and A. Renyi. On a new law of large numbers. J. Anal. Math. 2 3 , 103-111 (1970).
17. P. M. Frank. Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy - a survey and some new results. Automatica 26, 459-474 (1991).
18. D. Iglehart. Extreme values for the GI/G/1 queue. Ann. Math. Statist. 4 3 , 627-635 (1972).
19. S. Karlin, A. Dembo and T. Kawabata. Statistical composition of high scoring segments from molecular sequences. Ann. Statist. 18, 571-581 (1990).
20. T. L. Lai. Gaussian processes, moving averages and quick detection problems.
Ann. Probab. 1, 825-837 (1973).
21. T. L. Lai. Summability methods for independent, identically distributed ran- dom variables. Proc. Amer. Math. Soc. 45, 253-261 (1974a).
22. T. L. Lai. Limit theorems for delayed sums. Ann. Probab. 2, 432-440 (1974b).
23. T. L. Lai. Sequential change-point detection in quality control and dynamical systems (with discussion). J. Roy. Statist. Soc. Ser. B 57, 613-658 (1995).
24. T. L. Lai. Information bounds and quick detection of parameter changes in stochastic systems. IEEE Trans. Information Theory 44, 2917-2929 (1998).
25. T. L. Lai. Sequential multiple hypothesis testing and efficient fault detection- isolation in stochastic systems. IEEE Trans. Information Theory 46 595-608 (2000).
26. T. L. Lai and J. Z. Shan. Efficient recursive algorithms for detection of abrupt changes in signals and control systems. IEEE Trans. Automatic Control 44, 952-966 (1999).
27. G. Lorden. Procedures for reacting to a change in distribution. Ann. Math.
Statist. 42, 1897-1908 (1971).
28. G. Moustakides. Optimal procedures for detecting changes in distributions.
Ann. Statist. 14, 1379-1387 (1986).
29. I. V. Nikiforov. New optimal approach to global positioning sys- tem/differential global positioning system integrity monitoring. J. Guidance, Control & Dynamics 19, 1023-1033 (1996).
30. L. Shepp. A limit law concerning moving averages. Ann. Math. Statist. 35, 424-428 (1964).
31. A. N. Shiryayev. Optimal Stopping Rules. Springer-Verlag, New York, (1978).
32. D. Siegmund. Tail probabilities for the maxima of some random fields. Ann.
Probab. 16, 487-501 (1988).
33. D. Siegmund and E. S. Venkatraman. Using the generalized likelihood ratio statistics for sequential detection of a change-point. Ann. Statist. 23, 255-271 (1995).
34. D. Siegmund and K. J. Worsley. Testing for a signal with unknown location and scale in a stationary gaussian random field. Ann. Statist, 23, 608-639, (1995).
35. V. Strassen. A converse to the law of the iterated logarithm. Z. Wahrschein.
4, 265-268 (1996).
36. Q. Zhang, M. Basseville and A. Benveniste. Early warning of slight changes in systems and plants with application to condition based maintenance. Au- tomatica 30, 95-114 (1994).
PROCESSES*
LIMING WU
Laboratoire de Math. Appl. CNRS-UMR 6620, Universite Blaise Pascal, 63177 Aubiere, France. Email: Li-Ming.Wu@math.univ-bpclermont.fr
and
Department of Math., Wuhan University, 430072 Hubei, China
Let (xn = J^t^-oo ai-n^j) be the moving average process, where (£n)nez is a sequence of Revalued centered i.i.d.r.v. such that E e5^0' < +oo for some <S > 0.
Under the assumption that the spectral density function of X is continuous, we establish the process-level large deviation principle of X and t h e large deviations for empirical variance of X, and we identify their rate functions. Large deviations for empirical covariance and for empirical spectral measure are also considered under some stronger integrability condition on £o- Our main tools are some improved versions of the approximation lemma in the large deviation theory and an a priori estimation about the quadratic functional of X.
M S C 2000 Subject Classification: 60F10; 60G10; 60G15.
K e y Words: large deviations; moving average processes; Gaussian pro- cesses; spectral measures.
1. I n t r o d u c t i o n
Let (£n)nez be a sequence of Revalued centered square integrable i.i.d.r.v.
and (a„)„ez a sequence of real numbers such that
^ | a „ |2< + o o . (1)
nSZ
' T h i s work is partially supported by the Yangtze professorship of Wuhan University 15
Consider the moving average process
+ 0 0 4-00
Xn := ^2 aj-ntj = ^2 a&n+i> V n e Z- (2)
j = — 0 0 j=—oo
(Condition (1) is necessary and sufficient for the a.s. convergence or con- vergence in law of the series (2).) It appears very often in filtrage, time series analysis and engineering.
The classic limit theorems such as the central limit theorem, the law of iterated logarithm about moving average processes are a traditional subject in probability theory and its applications, see 14, 12. For example, the minimal condition for the central limit theorem of Sn := £fc=i Xk 1S (s e e 12, Corollary 5.2)
+ 0 0
g[ff) := V^ anem8 is continuous at 0 = 0 . (3)
n = —00
The main purpose of this paper is to investigate the large deviations of mov- ing average processes. Let us recall several known results which motivate this work.
I) About stationary Gaussian processes. When (£n) is a sequence of real Gaussian i.i.d.r.v. of law7V(0,1), X = (Xn) is a stationary Gaussian process, and inversely any real Gaussian stationary process (Xn) with a square integrable spectral density function can be represented as (2). In a pioneering work 10(1985), Donsker and Varadhan established the process- level (or level-3) large deviation principle (in short: LDP) of the process- level occupation measures
fe=i
where S. denotes the Dirac point measure. Their conditions for that LDP are
the spectral density function f(9) = \g{9)\2 is continuous on R; (5) and
k > g / ( 0 ) d 0 > - o o . (6) /
J — -
- 7 T
Moreover they obtained an explicit expression of the rate function governing the LDP of (Rn).
Bryc and Dembo 3(1995) established this same LDP, only under condi- tion (5). Especially they show that the continuity condition (5) is almost the best: if f(6) is bounded on [—n, 7r] but is discontinuous at 6 = 0, that LDP could fail. The price paid for this extension is that the beautiful expression of the rate function in Donsker-Varadhan 10 is lost.
Some words about the main ideas of Donsker-Varadhan10 and Bryc and Dembo3:
1) the first remark is: when (an) is of finite range, i.e., an = 0 for all \n\ > N for some N S N, this LDP is a consequence of the Donsker- Varadhan process level LDP for the i.i.d. sequence (£n)(9> 1975);
2) to show the LDP for (Rn) for general (an), it consists to approximate X by Fejer approximation
X ^ = E f 1 - 1 ) a ; W ( ? )
j=-N ^ '
The key for this approximation to be successful is the following a priori estimation under (5), due to 10(Lemma 2.4): there exists e(N) —> 0 (as N —> co) such that if 0 < A < 2e(N)' *^e n ^or a^ n — •*•'
— log E < e x p n
< - i l o g ( l - 2 A£( i V ) ) . (8)
A £ ( xn- X W ) '
. fc=i
That estimation as well as those in 3(Lemma 3.3) depends heavily on explicit calculations based on the Gaussian distribution (Toeplitz matrix), and we do not see how to extend their approach to more general distribu- tions.
The process-level LDP implies the LDP of £ £fc=i F(Xk, ••• , Xk+i) for all F bounded and continuous. But in practice functionals F are often un- bounded, such as the quadratic functional F(x) = x2, covariance functional F = XQXI etc. In this direction, Bryc and Dembo 2(1993) obtained large deviations of quadratic functionals of X = (Xn) under the weaker condition that / is bounded, extending and refining (8) in some sense.
Bercu, Gamboa and Rouault 1(1997) investigated the large deviations of Wn and of f* h(6)Wn(d6), where (Wn) is the empirical spectral measure given by
2
Wn(d6) := - Jk9 dB
fc=l
on the unit torus T identified as [—ir, ir], and h £ L°°(T). In their work,
fc=i
7r, 7r], and
the large deviation principle of f* h(9)Wn(d6) is characterized by some
necessary and sufficient condition on the distribution of eigenvalues of the related Toeplitz matrix; and that condition is verified in the white noise case Xn - fn for h continuous on T (moreover the functional type large deviations for W„ is obtained for the white noise case too).
For other references related to the large deviations of Gaussian pro- cesses, the reader is referred to 3, 1.
II) About general moving average processes. Burton and Dehling (1990) obtained the large deviations of the empirical means Sn/n :=
n Z)fc=i Xk under the following conditions
^ | a „ | < + o o (9)
n€Z
and
Eexp(A|£0|) < +oo, for all A > 0. (10) Jiang, Rao and Wang 13(1995) showed that the lower bound of large devia-
tions of Sn is valid without condition (10) and that the upper bound (with perhaps a different rate function) is true even if (10) holds for some A > 0.
Condition (9) is much stronger than (5), but condition (10) is much weaker than
Eexp(<5|£0|2) < +oo, for some <5 > 0 (11) satisfied by the Gaussian measure.
Recently Djellout and Guillin 8(2001) obtained the LDP of Sn by impos- ing the boundedness of £o, but with recompense, condition (9) is weakened as the minimal condition (3).
Let us notice that Sn is only a linear functional of (Xk). And the process level LDP, the large deviations of £ £= 1 F(Xn, • • • ,Xn+l) for a nonlinear functional F are left open in their works (such as the quadratic functionals, empirical covariance etc which are of important interests in practice).
The main objective of this paper is to establish the process level LDP, the large deviations of quadratic functionals or of empirical covariance and the LDP of empirical spectral measures etc, for general moving average processes under the assumptions (5) and (11). Our method, close to the pioneering work of Donsker and Varadhan 10(1985) and that of Bryc and Dembo 3(1995) but different from the method of Gartner-Ellis theorem, is based on some improved versions of the approximation lemma.
It is organized as follows. The main results are stated at the next Section 2. In Section 3 we establish the key a priori estimation extending (8). The last three sections are devoted to the proofs of the main results.