"ze Leung Lai Hailiang Yang Siu Pang Yung
PROBABILITY, FINANCE AND INSURANCE Proceedings of a Workshop at the University of Hong Kong
PROBABILITY, FINANCE AND INSURANCE
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PROBABILITY, FINANCE AND INSURANCE Proceedings of a Workshop at the University of Hong Kong Hong Kong
15 - 17 July 2002
editors
Tze Leung Lai Stanford University, USA
Hailiang Yang Siu Pang Yung The University of Hong Kong, China
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PREFACE This volume contains about half of the papers that were presented at the Workshop on Probability with Applications to Finance and Insurance on July 15-17, 2002 at The University of Hong Kong. Thirty-one invited speakers from mainland China, Hong Kong, Taiwan, Singapore, Australia and the United States gave talks at the workshop, which brought together many leading researchers in probability theory, stochastic processes, mathematical finance and actuarial science. The participants discussed important current issues and exchanged ideas, and there was fruitful interaction between researchers from different fields. The workshop was supported by the Institute of Mathematical Research at The University of Hong Kong. We are particularly grateful to Professor Ngaiming Mok, the Institute's Director, for suggesting the idea of a workshop of this kind and for his generous support. The workshop was also sponsored by the Hong Kong Mathematical Society. We also thank Professor Eric Chang of the Department of Finance, Dr. Kai Wang Ng of the Department of Statistics and Actuarial Science, Professor Man Keung Siu and the staff of the Department of Mathematics for their help and support. We are deeply grateful to Professor Inchi Hu and Dr. Bing-Yi Jing of the Hong Kong University of Science and Technology and Professor Qi-Man Shao of the University of Oregon and the National University of Singapore for working closely with us on the Organizing Committee in planning and making arrangements for the workshop. We want to express our gratitude to the authors for their careful preparation of the manuscripts, which have made publication of this volume possible, and to Professors Gerold Alsmeyer, Beda Chan, Wai Ki Ching, Wai Keung Li, Qihe Tang and Lixing Zhu for their help in the refereeing process. Thanks are also due to Ms. Ada Lai and Mimi Lui for their valuable administrative and technical assistance in the preparation of the Proceedings.
T. L. Lai, H. Yang and S.P. Yung Stanford and Hong Kong
V
LIST OF PARTICIPANTS
Andrew Carverhill The University of Hong Kong Hong Kong
Eric C. Chang The University of Hong Kong Hong Kong
Kani Chen Hong Kong University of Science and Technology Hong Kong
Ngai-Hang Chan The Chinese University of Hong Kong Hong Kong
Sung Nok Chiu Hong Kong Baptist University Hong Kong
Kwok Pui Choi National University of Singapore Singapore
August Chow Office of the Commissioner of Insurance Hong Kong
Cheng-Der Fuh Institute of Statistical Science Academia Sinica Taipei
Fu Zhou Gong Institute of Applied Mathematics Chinese Academy of Sciences Beijing
Minggao Gu The Chinese University of Hong Kong Hong Kong
Inchi Hu Hong Kong University of Science and Technology Hong Kong
Yaozhong Hu University of Kansas Lawrence, KS 66045
vu
Hsien-Kuei Hwang Institute of Statistical Science Academia Sinica Taipei
Bing-Yi Jing Hong Kong University of Science and Technology Hong Kong
Yue-Kuen Kwok Hong Kong University of Science and Technology Hong Kong
Tze Leung Lai Stanford University Stanford, CA 94305
Wai Keung Li The University of Hong Kong Hong Kong
Szu-Lang Liao Banking and Financial Markets National Cheng-Chi University Taipei
Tiong Wee Lim National University of Singapore Singapore
Shiqing Ling Hong Kong University of Science and Technology Hong Kong
Ngaiming Mok The University of Hong Kong Hong Kong
Kai Wang Ng The University of Hong Kong Hong Kong
Qi-Man Shao University of Oregon Eugene OR 97403-1217
Elias Sai Wan Shiu University of Iowa Iowa City, Iowa 52242-1409
Man Keung Siu The University of Hong Kong Hong Kong
Chun Su University of Science and Technology of China Hefei
Qiying Wang Australian National University Canberra, ACT, 0200 Australia
Ching-Zong Wei Institute of Statistical Science Academia Sinica Taipei
Michael Wong The Chinese University of Hong Kong Hong Kong
Liming Wu Wuhan University Wuhan
Kai-Nan Xiang Hunan Normal University Changsha
Hailiang Yang The University of Hong Kong Hong Kong
Xiao-Guang Yang Institute of System Science Chinese Academy of Sciences Beijing
Yi Ching Yao Institute of Statistical Science Academia Sinica Taipei
Zhiliang Ying Columbia University New York, NY 10027
Siu Pang Yung The University of Hong Kong Hong Kong
Prank Zhang Morgan Stanley New York
Lixin Zhang Zhejiang University Hangzhou
Xunyu Zhou The Chinese University of Hong Kong Hong Kong
CONTENTS Preface
v
List of Participants
vi
Limit theorems for moving averages 1 Tze Leung LAI On large deviations for moving average processes 15 Liming WU Recent progress on self-normalized limit theorems 50 Qi-Man SHAO Limit theorems for independent self-normalized sums 69 Bing-Yi JING Phase changes in random recursive structures and algorithms 82 Hsien-Kuei HWANG Iterated random function system: convergence theorems 98 Cheng-Der FUH Asymptotic properties of adaptive designs via strong approximations .112 Li-Xin ZHANG Johnson-Mehl tessellations: asymptotics and inferences 136 Sung Nok CHIU Rapid simulation of correlated defaults and the valuation of basket default swaps 150 Zhifeng ZHANG, Kin PANG, Peter COTTON, Chak WONG and Shikhar RANJAN Optimal consumption and portfolio in a market where the volatility is driven by fractional Brownian motion 164 Yaozhong HU MLE for change-point in ARMA-GARCH models with a changing drift 174 Shiqing LING Dynamic protection with optimal withdrawal 195 Hans U. GERBER and Elias Sai Wan SHIU Ruin probability for a model under Markovian switching regime 206 Hailiang YANG and G. YIN Heavy-tailed distributions and their applications 218 Chun SU and Qihe TANG The insurance regulatory regime in Hong Kong (with an emphasis on the actuarial aspect) 237 August CHOW
ix
LIMIT THEOREMS FOR M O V I N G AVERAGES
TZE LEUNG LAI Stanford University, CA, USA E-mail:
[email protected] Moving averages have many important statistical applications, particularly in change-point problems and signal detection (for which the moving averages are taken over spatial domains). Herein we give a review of limit theorems for moving averages and describe some recent developments motivated by applications to signal detection and change-point problems.
1. Introduction Let £,£i,£2i-" be i.i.d. random variables, and let Sn = £f =1 £i, So = 0, Sn(k) = Sn — Sn_fe. The window size k of the moving average k~1Sn(k) can be time-invariant or can vary with n, yielding fc~15n(fcn) as the moving average. The special case kn = n reduces to the usual sample mean Sn/n, for which one has the Kolmogorov and Marcinkiewicz-Zygmund strong laws: n~1Sn —> 11 a.s. 4=> E\$\ < oo and E£ = n, rTl/vSn
-> 0 a.s. <s=^ £|£| p < co and E£ = 0
(l /?„ a.s.,
(4)
2
where b = (3a solves the equation e~ 1 / a = inf t >oe _ t 6 i?e t f . This is often called the Erdos-Renyi-Shepp strong law, as Shepp (1964) proved earlier the closely related result maxkn<j /3a a.s. Deheuvels et al. (1986) subsequently proved the following refinement: Let t = ta be the unique solution of (EXetX)/Eetx = /?„. Then p-Um (logfc n ) -1 { max Sj{kn) = p-lim (logfc„)"~1
oo. For the former, we also generalize fc-1Sn(fc) to weighted sums of the form S^LiCn-j^, in which c = (co,ci,...) € £2 (i.e., S g 0 c ? < oo), noting that fe_15n(fc) corresponds to the case Co = c\ = ... = Ck-i = 1/fc and c* = 0 for i > k. Analogous to (l)-(3), necessary and sufficient moment conditions for £ are given for these almost sure limits (or limsup,liminf) of the moving averages. We first consider the case of windows of fixed size k. Chow and Lai (1973) proved the following analogue of (2): For p > 1, n-^pSn(k)
-> 0 a.s. <s> E\£\p < oo -> 0 a.s. for all c ^ 0 with Yffc2 < oo,
•» n-VfYZcn-iti
S n (fc)/logn ->• 0 a.s. & Eem
(8)
< oo for all t > 0
<S> E^Cn-i^/logn -> 0 a.s. for all c ^ 0 with £g°cf < oo.
(9)
For fixed k, {Sn(k), n > k} is a fc-dependent stationary sequence; more generally, we expect Yn := £™c„_,£j to share the same almost sure (a.s.) limiting properties. In particular, Lai (1973) proved the following upper/lower class characterization of boundaries bn for Yn when the £ n are Gaussian, analogous to the case of i.i.d. standard normal Yn. T h e o r e m 2 . 1 . Suppose £ is standard normal, a2 := ££L0c? < oo and E g n c ? = O((logg)~ 2 ). Let bn be an eventually nondecreasing sequence of positive numbers. Then
f "
]
P< 2.°n-i^i
— bn infinitely often > oo
= 0 (or 1) if Y2bnle~bl/{2°2)
< (°r
=)oo.
n=l
We next consider the case k = kn —> oo as n —» oo. A continuous function / : (0, oo) —> (0, oo) is called self-neglecting if f(x) = o(x) and f{x + tf(x))/f(x) —> 1 as x —> oo, for all t e R. It is said to be of bounded increase if there exist A > 1,C, a and £o such that f(Xx)/f(x) < C\a for all 1 < A < A and x > x0. Suppose / is continuous, increasing and self-neglecting and the derivative of f~l is of bounded increase. Bingham and Goldie (1988) proved the following analogue of (1) for moving averages with window sizes kn = f(n), assuming that lim^-,,^ f(x) — oo: Sn(f(n))/f{n)
-> fi a.s. & E/ _ 1 (|£|) < oo and E£ = \x.
(10)
4
They also showed that this result is closely related to Riesz summability of {£ n }. Bingham and Rogers (1991) gave a survey of this and other summability methods for i.i.d. random variables, including iterated-logarithm-type results and necessary and sufficient moment conditions for Cesaro, Abel, Euler and Borel summability considered by Lai (1974a). The following analogue of (3) for Sn(kn) was proved by Lai (1974b) when bna with 0 < a < 1: limsup„^ 0 0 5 n (fc n )/(fc r i logn) 1 / 2 = {2(1 - a ) } 1 / 2 ^ and liminf = -{2(1 - a)}1'"2a a.s. E£2 = a2 and Em2/a(log+
fcl
+ l)'1/a}
< oo.
(11)
Under additional moment conditions, Csorgo and Revesz (1981, Chapter 3) showed that for fcn/logn —* oo, limsup„^ TO ( i < f e < m a x < = a a.s.
) -1 (x).
(16)
9
In particular, when d = 1 and the Xt are i.i.d. standard normal, g(x) = x2/2, for which Siegmund and Venkatraman (1995) showed that as c —> oo, (,\fc e"cTc has a limiting exponential distribution with mean 1,
(17)
where C = ^ 1 / 2 Jo°° xv2{x)dx and v(x) = 2 : z r 2 e x p { - 2 £ f r T 1 $(—Xy/n/2)} for x > 0, in which $ denotes the standard normal distribution. Note that (17) is a special case of (14) with r = q = 1. The asymptotic theory in Chan and Lai (2003a,b) concerning (6) and (7) unifies these previous results and also leads to definitive solutions of a variety of change-point detection problems, giving in particular (i) the extension of i.i.d. to Markov-dependent & (so that more general stochastic systems can be treated), and (ii) suitable choice of g and Jn or J{c) in (6) or (7) to achieve both statistical and computational efficiency. A unified approach in Chan and Lai (2003a) to derive (14) and (15) is based on integrating saddlepoint approximations for Markov random walks
7
with respect to certain measures over some tubular neighborhood of an extremal g-dimensional manifold in R d , which incorporates both the critical temporal and spatial components of the problem, yielding an asymptotic formula for the following boundary crossing probability: P { max kg(Sn(k)/k)
> c for some n < /3c}
5c c}. Then the weak convergence property implies that for fixed c, T 7 / 7 converges in distribution under PM/V=r toT M (C) := inf{t: max 0 6 l 7 :
max
5j(fc)Vr_15n(fc)/2fc > c},
(20)
b\i 0. The preceding asymptotic approach has been called "local" because it is based on weak convergence of T 7 / 7 or T 7 / 7 under P^ to the same limiting distribution as that in the canonical setting of independent normal V~l/2£t, when A7 is of the order X/y/j in the CUSUM rule or when the windows are of the form 617 < k < 627 in (20). Such choice of window size (or A7) makes T 7 (or T 7 ) very inefficient for detecting changes that are considerably larger than the 0 ( 7 - 1 / 2 ) order of magnitude for "local" changes. Lai and Shan (1999) proposed another approach based on moderate deviations theory (instead of weak convergence approximations) to analyze boundary crossing probabilities associated with detection rules of the general form T* = inf{n > M : max £%(k)V^Sn{k)/2
> c},
(21)
where c ~ log7 (instead of bounded c in the asymptotic local approach), JM = {Mj • J € J(c)}, J(c) = { 1 , . . . , M } U {[<JM] : 1 < j < J*},UJ >
9
1, M ~ ac, j * = min{j : [wJM] > 7} ~ c/logw, M/c —• 00 (to ensure that the sums J2"_fc & become approximately Gaussian in the moderate deviations sense), and Vn,k is an estimate of the asymptotic covariance matrix of Sn(k) under P0. Note that V_1/k in (20) corresponds to the inverse of Vn,k = kV. In many applications V is unknown and needs to be estimated. Even when V is known, using an estimate Vn,k instead of kV offers the flexibility of making adjustments for non-normality in treating (21) under PQ as if it were a window-limited GLR rule with independent standard normal V - 1 / , 2 £ti where the £t are actually dependent random vectors with unknown distributions. Grouping the observations into batches as in (21) arises naturally in quality control applications, in which samples of size M are taken at regular intervals of time. The moderate deviation approximations to boundary crossing probabilities associated with (21) involve corresponding probabilities for Gaussian processes. When the £t are i.i.d. random variables, the associated Gaussian process is Brownian motion B(t) and the increments of Sn are independent and approximately normal in the moderate deviations sense. In this case, the following moderate deviations counterpart of (18) holds, as has recently been shown by Chan and Lai (2003b): For 0 < a\ < 02 < a, P{{h - ti)~ 1 / 2 [B(t 2 ) - B(h)} > c for some 0 a n ( i for a > 0 and positive integers m, {c[Xc(t + akAc) - Xc(t)} :0 {Wt(ak)
:0 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 j a such that 7 a —> 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} < iV 0 ( 7 )V(c), 0 R+ such that /(||r||) = 0(e _ ll r H P ) for some p > 0 and for all 7 > 0 and c sufficiently large, (A5)
P{Xc(t)
> c - 7/c, Xc(t + uA c ) > c - 7/c} < rl>(c - 7/c)/(||u||)
uniformly in t and t + uAc belonging to [D]$. Note that (Al) says that the "moderate deviation" event {Xc(t) > c — y/c} has probability like that of a standard normal. Whereas this refers to the marginal distribution of Xc(t), the joint distribution is assumed in (A2) to be asymptotically normal in the sense of weak convergence for local increments conditioned on Xc(t) = c—y/c. Note that the same a, L(-) and rt(-) appear in (C) and the mean and covariance functions (23) of the limiting Gaussian field Wt(-) in (A2). In fact, if Xc = X is a Gaussian field satisfying condition (C), then (A2) holds; see Corollary 2.7 of Chan and Lai (2003c). Assumptions (A3)-(A5) are mild technical conditions under which the probability of sup u € / t KA Xc(u) exceeding c can be computed via (Al) and (A2), yielding the following asymptotic formulas. Theorem 5.1. (i) Let K > 0. Assume (C) and (Al)-(A4). P{
sup
Xe(u) > c} ~ V(c)[l + HK(t)]
Then (25)
«6fl,K4c
uniformly overt G [D}$, where HK(t) = /0°° e 3 'P{sup 0 < u . y} dy is finite and uniformly continuous int £ [D}$. (ii) Assume (C) and (Al)-(A5). Then H{t) = lim K -_ f0O ii'- d HK-(t) exists and is uniformly continuous and bounded below on D. Moreover, as c —» 00 and £c —> 00 such that lc = o(A~1), P{
sup ueit,tcAc
Xc{u) > c} ~ edci(>(c)H(t),
(26)
12
P{
sup uElt,ecAc
Xc{u) > c,
sup
Xc(v) >c} = o(^V(c)),
(27)
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, asymptotic boundary crossing "density" A~dip(c)H(t) of Xc at tegrating this "density" over D, or more precisely, by summing the "tiles" It,ecAa of D and applying (27) together with the fact bounded and Jordan measurable, it follows that
yields an t. By in(26) over that D is
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 detection 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[ nt j 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 approach 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 Multivariate 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 almostsure 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 boundary crossing probabilities of Markov random walks. Ann. Appl. Probab. 13, 394-428 (2003a).
13
8. H. P. Chan and T. L. Lai. Moderate deviation approximations for the increments 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 random 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 Statistics. 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. 43, 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 random 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 detectionisolation 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.
14
Ann. Statist. 14, 1379-1387 (1986). 29. I. V. Nikiforov. New optimal approach to global positioning system/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. Automatica 30, 95-114 (1994).
ON LARGE DEVIATIONS FOR MOVING AVERAGE PROCESSES*
LIMING WU Laboratoire de Math. Appl. CNRS-UMR 6620, Universite Blaise Pascal, Aubiere, France. Email:
[email protected] and Department of Math., Wuhan University, 430072 Hubei, China
Let (xn
= J^t^-oo
a
i-n^j)
63177
be the moving average process, where (£n)nez is
a sequence of Revalued centered i.i.d.r.v. such that E e 5 ^ 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 the 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. Key Words: large deviations; moving average processes; Gaussian processes; spectral measures.
1. Introduction Let (£n)nez be a sequence of Revalued centered square integrable i.i.d.r.v. and (a„)„ e z a sequence of real numbers such that ^|a„|2 0 (as N —> co) such that if 0 < A < 2e(N)' *^ e n ^ or a ^ n — •*•' — log E < e x p A £ ( x n - X W ) ' n . fc=i
0. Condition (9) is much stronger than (5), but condition (10) is much weaker than Eexp( [0, +oo] is inf-compact, i.e., VX > 0, the level set [I < L] is compact in E (or I is a good rate function); (ii) (lower large deviations: LLD in short) for each open subset O C E, hminf —- T log/i„(0) > - inf I(x); n-»oo A(n) xeo (Hi) (upper large deviations: ULD in short) for each closed subset C C E, limsup —r-rlog/u n (C) < - inf n-xx,
A(n)
I(x).
xGC
(Convention: inf 0 := +00,). In that case we write (/zn) £ LDP(E,X(n),I) simply, and when \xn = the law of r.v. Xn, we write (Xn) S LDP(E,X(n),I) directly. We begin by improving a well known approximation lemma in Dembo and Zeitouni 6 , Thm.4.2.16(1998): Theorem 2.1. If (Zn '),(Zn) ity space (Cl^tF) such that (i) for each N, (ziN)\
e
are E-valued r.v. defined on some probabilLDP(E,\{n)j(N));
(H) {Z{nN)) is an exponential good approximation of (Zn), i.e., V5 > 0, lim l i m s u p - i - l o g P ( p ( Z W , Z „ ) > d) = - c o . JV-»oo n ^ o o
A(n)
V
/
20
Then (Zn) G LDP(E,X(n),I)
where
I(x) :=supliminf inf / W = s u p U m s u p inf I{N), 0 N-KX> B(x,6)
s>0 JV->oo
(12)
B(x,8)
where B(x, 5) = {y G E\ p(x, y) < 6} is the ball centered at x with radius 6. The main difference from 6 , Thm.4.2.16 (1998) is: their technical assumption (b) (including the inf-compactness of I) is dropped, and the infcompactness of I becomes now a consequence. This will be very useful for our purpose. The classical Contraction Principle says: if (/i n ) G LDP(E,\(n),I) and if / : E —» F is continuous where F is another Polish space, then (Mn o / - 1 ) G LDP(F, A(n), If) where If(y) := inf {I(x)\x £ E; f(x) = y} , Vy e F. But in many applications (the case in this paper), / is not continuous. The following result treats this discontinuous case: Theorem 2.2. (Generalized Contraction Principle) Let E, F be two Polish spaces and dF a metric compatible with the topology of F. Given a family of probability measures (/i„) on E such that (/xn) € LDP(E,X(n),I), and a measurable application f : E —> F. If there is a sequence of continuous mappings fN : E —• F such that, lim limsup—^-log/i„ (dF(fN,f) AT—oo n ^ o o
> S) = -oo,V 0,
A(n)
then there is a continuous function f :[I < +oo] —> F such that sup dF(fN(x)J)
—^0{asN-*
oo), VL > 0;
(13)
< +oo; / » = y) , Vy G F.
(14)
x€[I) = ±Y/5xm=Ono($Wy\
Rn=Ono$-1,P-a.s.
(22)
fe=i
Since $ w is continuous from f2 to W, the application Q —> Q ° ( $ M ) - i is continuous from Mj(f2) to M ^ W ) . Assume that we could prove that for some appropriate metric dw on Mi(W) (compatible with the weak convergence topology), V 0,
^JTJ^^0gF{dW
^E^>(^)^E**(r^)J >*J =-00.
(23) Then by Theorem 2.2, we have the following (except the identification of the rate function which requires some more efforts).
23
Theorem 2.3. Assume (16) and (17). Then there is an application $ : f2 —> W such that $ 0 ( T ' W ) = $i{u), V/ G Z,w G fi; 2
sup Q : i f ( Q ) < L / $ ( W - $( d
