GENERAL ASYMPTOTIC BAYESIAN THEORY OF QUICKEST CHANGE DETECTION. Alexander G. Tartakovsky and Venugopal V. Veeravalli

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1 Fina Version Submitted to the Theory of Probabiity and Its Appications, May 1, GENERAL ASYMPTOTIC BAYESIAN THEORY OF QUICKEST CHANGE DETECTION Aexander G. Tartakovsky and Venugopa V. Veeravai ABSTRACT. The optima detection procedure for detecting changes in independent and identicay distributed i.i.d. sequences in a Bayesian setting was derived by Shiryaev in the nineteen sixties. However, the anaysis of the performance of this procedure in terms of the average detection deay and fase aarm probabiity has been an open probem. In this paper, we deveop a genera asymptotic change-point detection theory that is not imited to a restrictive i.i.d. assumption. In particuar, we investigate the performance of the Shiryaev procedure for genera discrete-time stochastic modes in the asymptotic setting where the fase aarm probabiity approaches zero. We show that the Shiryaev procedure is asymptoticay optima in the genera non-i.i.d. case under mid conditions. We aso show that the two popuar non-bayesian detection procedures, namey the Page and the Shiryaev-Roberts-Poak procedures, are generay not optima even asymptoticay under the Bayesian criterion. The resuts of this study are shown to be especiay important in studying the asymptotics of decentraized change detection procedures. Keywords and Phrases: Change-point detection, sequentia detection, asymptotic optimaity, noninear renewa theory. 1. Introduction The probem of detecting abrupt changes in stochastic processes arises in a variety of appications incuding biomedica signa processing, quaity contro engineering, finance, ink faiure detection in communication networks, intrusion detection in computer systems, and target detection in surveiance systems [1, 4, 12, 19, 30]. A typica such probem is one of target detection in mutisensor systems radar, infrared, sonar, etc. [30, 35, 39], where the target appears randomy at an unknown time. The goa is to detect the target as quicky as possibe, whie maintaining the fase aarm rate at a given eve. Another appication area is intrusion detection in distributed computer networks [4, 12, 39]. Large scae attacks, such as denia-of-service attacks, occur at unknown points in time and need to be detected in the eary stages by observing abrupt changes in the network traffic. The design of the quickest change detection procedures usuay invoves optimizing the tradeoff between two kinds of performance measures, one being a measure of detection deay and the other being a measure of the frequency of fase aarms. There are two standard mathematica formuations for the optimum tradeoff probem. The first of these is a minimax formuation proposed by Lorden [17] and Poak [21], in which the goa is to minimize the worst-case deay subject to a ower bound on the mean time between fase aarms. The second is a Bayesian formuation, proposed by Shiryaev [25]-[27], in which the change point is assumed to have a geometric prior distribution, and the goa is to minimize the expected deay subject to an upper bound on fase aarm probabiity. The asymptotic performance of various change-point detection procedures is we understood in the minimax context for both the discrete and continuous-time cases see [1],[3],[7],[9], [17],[18],[20],[21], [22],[23],[28],[30],[31],[32], [34],[39],[43],[44]. However, there has been itte previous work on the asymptotics of Bayesian procedures. The exception is the work by Lai [16] in which the asymptotic properties of Page s cumuative sum CUSUM procedure were studied in a Bayesian as we as minimax context for genera stochastic modes. See aso Beibe [3] for continuous-time Brownian motion and Borovkov [5], Lemma 5 for i.i.d. data modes. Our goa is to provide a genera Bayesian asymptotic theory for change point detection. The paper is organized as foows. In Section 2, we formuate the probem. In Section 3, we study the behavior of the Shiryaev detection procedure for genera, non-i.i.d. data modes, and prove that it is asymptoticay optima under mid conditions when the fase aarm probabiity goes to zero. We show that this procedure is asymptoticay optima not ony with respect to the average detection deay, but that it is aso uniformy asymptoticay optima in the sense of minimizing the conditiona expected deay for every change point. Moreover, we study the behavior of higher moments of the detection deay and show that under certain genera conditions the Shiryaev procedure minimizes moments of the detection deay up to the given order. In Section 4, we find the asymptotic operating characteristics of the Shiryaev change detection procedure in the i.i.d. case when the fase 1

2 2 A. TARTAKOVSKY AND V. VEERAVALLI aarm probabiity goes to zero. The use of noninear renewa theory aows us to obtain sharp asymptotic approximations for the fase aarm probabiity and the average detection deay up to vanishing terms. In Section 5, we anayze the asymptotic performance of other we-known change detection procedures Page s procedure and the Shiryaev-Roberts-Poak procedure in the Bayesian framework. The resuts of this section aow us to concude that, whie being optima in the minimax context, these procedures may ose their optimaity property even asymptoticay with respect to the Bayesian criterion depending on the structure of the prior distribution. In Section 6, we consider an exampe of detecting a change in the mean vaue of an autoregressive process that iustrates genera resuts. In Section 7, we consider two additiona exampes reated to detecting changes in distributed muti-sensor systems. We study the impications of the asymptotic resuts in decentraized quickest change detection probems assuming that sensors send quantized versions of their observations to a fusion center centra processor where the change detection is performed based on a the sensor messages. Finay, in Section 8, we concude the paper by giving severa remarks. The resuts of this paper have been presented in part at the Internationa Symposium on Information Theory [36] and the Information Theory Workshop [37] in 2002, and at the Sixth Internationa Conference on Information Fusion [38] in Probem Formuation In the conventiona setting of the change-point detection probem one assumes that the observed random variabes X 1, X 2,... are i.i.d., unti a change occurs at an unknown point in time λ, λ 1, 2,.... After the change occurs, the observations are again i.i.d. but with another distribution. In other words, conditioned on λ = k, the observations X 1, X 2,... are independent with X n f 0 for n < k and X n f 1 for n k, where f 0 x and f 1 x are, respectivey, the pre-change and post-change probabiity density functions pdf with respect to a sigma-finite measure µ. For the sake of brevity, in what foows, this case wi be referred to as the i.i.d. case. In a Bayesian setting, the change point λ is assumed to be random with prior probabiity distribution π k = Pλ = k, k = 0, 1, 2,.... The goa is to detect the change as soon as possibe after it occurs, subject to the fase aarm probabiity constraints. In mathematica terms, a sequentia detection procedure is identified with a stopping time τ for an observed sequence X n n 1, i.e. τ is an extended integer-vaued random variabe, such that the event τ n beongs to the sigma-agebra Fn X = σx 1,..., X n. A fase aarm is raised whenever the detection is decared before the change occurs, i.e. when τ < λ. A good detection procedure shoud guarantee a stochasticay sma detection ag τ λ provided that there is no fase aarm i.e. τ λ, whie the rate of fase aarms shoud be ow. Let P k and E k denote the probabiity measure and the corresponding expectation when the change occurs at time λ = k. In what foows, P π stands for the average probabiity measure which is defined as P π Ω = k=0 π kp k Ω, and E π denotes the expectation with respect to P π. In the Bayesian setting, a reasonabe measure of the detection ag is the average detection deay ADD 2.1 ADDτ = E π τ λ τ λ = Eπ τ λ + P π τ λ = 1 P π τ λ and the fase aarm rate can be measured by the probabiity of fase aarm 2.2 PFAτ = P π τ < λ = π k P k τ < k. π k P k τ ke k τ k τ k, Here and henceforth, we use the traditiona notation x + = max0, x for the positive part of x. An optima Bayesian detection procedure is a procedure for which ADD is minimized, whie PFA is constrained to be beow a given usuay sma eve α, α 0, 1. Specificay, define the cass of change-point detection procedures α = τ : PFAτ α for which the fase aarm probabiity does not exceed the predefined number α. The optima change-point detection procedure is described by the stopping time ν = arg inf ADDτ. τ α k=0

3 ASYMPTOTIC THEORY OF QUICKEST CHANGE DETECTION 3 Obviousy, if PFAν = α, then ν aso minimizes E π τ λ +. Let X n = X 1,..., X n denote the concatenation of the first n observations, et Fn X = σx n be a sigmaagebra generated by X n, and et p n = Pλ n Fn X be the posterior probabiity that the change occurred before time n. For the i.i.d. case, Shiryaev [25]-[27] proved that if the distribution of the change point is geometric, i.e., Pλ = 0 = π 0, π k = 1 π 0 ρ1 ρ k 1, k 1 0 < ρ < 1, 0 π 0 < 1, then the optima detection procedure is the one that raises an aarm at the first time such that the posterior probabiity p n exceeds a threshod A, i.e νa = inf n 1 : p n A, where the threshod A = A α shoud be chosen so that PFAνA = α. A generaization of this resut for an arbitrary prior distribution has been stated in Borovkov [5], abeit without proof see Theorem 8 in [5]. However, except for the case of detecting the change in the drift of the Wiener process observed in continuous time, it is difficut to find a threshod that provides an exact match to the given PFA. Aso, there are no resuts reated to the ADD evauation of this optima procedure, again, except for the continuous-time Wiener process [27] with exponentia prior distribution, and for i.i.d. data modes with a geometric prior distribution when ρ 0 see Lemma 5 in [5]. Whie the exact match of the fase aarm probabiity is reated to the estimation of the overshoot in the stopping rue 2.3, and for this reason is probematic, a simpe upper bound, which ignores overshoot, can be obtained. Indeed, since P π νa < λ = E π 1 p νa and 1 pνa 1 A on νa <, we obtain that the PFA defined in 2.2 obeys the inequaity 2.4 PFAνA 1 A. It foows that setting A = A α = 1 α guarantees the inequaity PFAνA α α. Note that inequaity 2.4 hods true for arbitrary proper, not necessariy geometric, prior distributions. More generay, assume that observations are non-i.i.d. in the pre-change and post-change modes. Specificay, et P stand for the probabiity measure under which the conditiona density of X n given X n 1 = X 1,..., X n 1 is f 0,n X n X n 1 for every n 1 i.e. λ =. Furthermore, for any λ = k, 1 k <, et P k stand for the probabiity measure under which the conditiona density of X n is f 0,n X n X n 1 if n k 1 and is f 1,n X n X n 1 if n k. Therefore, if the change occurs at time λ = k, then the conditiona density of the k-th observation changes from f 0,k X k X k 1 to f 1,k X k X k 1. The og-ikeihood ratio LLR for the hypotheses that the change occurred at the point λ = k and at λ = no change at a is 2.5 Z k n := og dp k dp X n = n t=k og f 1,tX t X t 1 f 0,t X t X t 1, k n. In what foows, we wi use the convention that before the observations become avaiabe i.e. for n = 0, Z0 0 = og[f 1,0X 0 /f 0,0 X 0 ] = 0 amost everywhere. In the rest of the paper, we wi consider prior distributions π k = Pλ = k concentrated on nonnegative integers. The foowing two casses of prior distributions wi be covered: distributions with exponentia tai for which og Pλ k im = d, d > 0; k k and prior distributions with heavy tais for which og Pλ k im = 0. k k The first cass wi be denoted by Ed, and the second cass by H. Ceary, the geometric prior distribution with the parameter ρ, 2.8 π k = π 0 1 k=0 + 1 π 0 ρ1 ρ k 1 1 k 1, 0 < ρ < 1, 0 π 0 < 1, 1 Hereafter we use the convention that inf =, i.e. νa = if no such n exists.

4 4 A. TARTAKOVSKY AND V. VEERAVALLI beongs to the cass Ed with d = og1 ρ. Here and henceforth, 1 A denotes an indicator of the set A. Note that a more genera case where a fixed d is repaced with the vaue of d α that depends on α and vanishes when α 0 can aso be handed in a simiar way. This more genera case has been considered by Lai [16]. However, this sight generaization does not have substantia impact on practica appications. For n 0, define the ikeihood ratio of the hypotheses H 1 : λ n and H 0 : λ > n Λ n := dpxn H 1 dpx n H 0 = n k=0 π n k i=k f 1,iX i X i 1 k 1 i=1 f 0,iX i X i 1 k=n+1 π n k i=1 f 0,iX i X i 1, where f 1,0 X 0 /f 0,0 X 0 = 1 amost everywhere by the above convention. Reca that p n = Pλ n Fn X denotes the posterior probabiity of the fact that the change occurred before time n. Write Π n = Pλ > n. It is easiy verified that 2.9 Λ n = Λ 0 + Π 1 n n π k e Zk n and Λ n = p n /1 p n, n 1, where Λ 0 = π 0 /1 π 0. Therefore, the Shiryaev stopping rue given in 2.3 can be written in the foowing form that is more convenient for asymptotic study ν B = inf n 1 : Λ n B, B = A/1 A, where B > π 0 /1 π 0. Evidenty, inequaity 2.4 hods in the genera, non-i.i.d. case too. Consequenty, 2.11 B α = 1 α/α impies ν Bα α assuming that α < 1 π 0. It is worth noting that whie the Shiryaev procedure 2.10 is optima in the i.i.d. case if B is chosen so that PFAν B = α, it may not be optima in the non-i.i.d. scenario even if we can set the threshod to meet the PFA constraint exacty. In fact, the properties of the Shiryaev procedure in the non-i.i.d. scenario have not been investigated previousy. In addition to the Bayesian ADD defined in 2.1, we wi aso be interested in the behavior of the conditiona ADD CADD for the fixed change point λ = k, which is defined by CADD k τ = E k τ k τ k, k = 1, 2... as we as higher moments of the detection deay E k τ k m τ k and E π τ λ m τ λ for m > 1. In the next section, we study the operating characteristics of the Bayesian procedure 2.10 for sma PFA α 0 in the genera, non-i.i.d. case. In Section 4, these resuts wi be speciaized to the i.i.d. scenario. 3. Asymptotic Operating Characteristics of the Detection Procedure ν B in a Non-i.i.d. Case As we mentioned above, in genera, the Shiryaev procedure ν B is not optima even if one is abe to chose the threshod B in such a way that PFAB = α. However, beow we show that this procedure with B = B α = 1 α/α is asymptoticay optima for sma PFA under some mid conditions. We wi show that in the asymptotic setting, ν Bα minimizes not ony the ADD, but aso CADD k for a k 1. Furthermore, under certain genera conditions this procedure minimizes higher moments of the detection deay up to the given order Asymptotic ower bounds for moments of the detection deay. We begin by estabishing asymptotic ower bounds for moments of the detection deay, in particuar for ADD and CADD of any procedure in the cass α. Later on these bounds wi be used to obtain asymptotic optimaity resuts. As we wi see, the derivation of the ower bounds is based on the appication of the Chebyshev inequaity that invoves certain probabiities, which are shown to go to 0 when α 0. We start with the study of the behavior of these probabiities. Let q be a positive finite number and define L α = L α q d = og α /q d, γ ε,α τ = P π λ τ < λ + 1 εl α, γ k ε,ατ = P k k τ < k + 1 εl α, 0 < ε < 1,

5 ASYMPTOTIC THEORY OF QUICKEST CHANGE DETECTION 5 where q d = q + d in the case of prior distributions with exponentia right tai, π Ed, and q d = q 0 = q in the case of heavy-taied prior distributions π H. The significance of the number q shoud be expained in more detai. We do not assume any particuar mode for the observations, and as a resut, there is no structure of the LLR process. We hence have to impose some conditions on the behavior of the LLR process at east for arge n. It is natura to assume that there exists a positive finite number q = qf 1, f 0 such that Z k n/n k converges amost surey to q, i.e n Zk P k a.s. k+n q for every k <. n This is aways true for i.i.d. data modes, in which case q = Df 1, f 0 = E 1 Z1 1 is the Kuback-Leiber information number. It turns out that the a.s. convergence condition 3.1 is sufficient for obtaining ower bounds for a positive moments of the detection deay but not necessary. In fact, the key condition 3.2 in Lemma 1 and Theorem 1 hods whenever Zn/n k k converges amost surey to the number q. Therefore, this number is of paramount importance in the genera change-point detection theory. Note that, since τ < k beongs to the sigma-fied Fk 1 X, P kτ < k = P τ < k, and hence, PFAτ = π k P τ < k. The foowing fundamenta emma wi be repeatedy used to derive ower bounds for the performance indices. LEMMA 1. Let Zn k be defined as in 2.5 and assume that for some q > P k M max 0 n<m Zk k+n 1 + εq 0 for a ε > 0 and k 1. M Then, for a 0 < ε < 1 and k 1, 3.3 im and for a 0 < ε < 1, 3.4 im sup α 0 τ α sup α 0 τ α γ k ε,ατ = 0, γ ε,α τ = 0. PROOF. Changing the measure P P k, we obtain that for any C > 0 and ε 0, 1 P k τ < k + 1 εl α = E k 1 k τ<k+1 εlαe Zk τ E k 1 k τ<k+1 εlα,zτ k<c e Zk τ 3.5 e C P k τ < k + 1 εl α, max Zn k < C k n<k+1 εl α [ ] e C P k k τ < k + 1 εl α P k max Zk+n k C, 0 n<1 εl α where the ast inequaity foows triviay from the fact that for any events A and B with B c being a compement to B, PA B PA PB c. Setting C = 1 ε 2 ql α, we obtain 3.6 Write γ k ε,ατ e 1 ε2 ql α P k τ < k + 1 εl α + P k max Zk+n k 1 0 n<1 εl ε2 ql α. α p k α, ε = e 1 ε2 ql α P k τ < k + 1 εl α

6 6 A. TARTAKOVSKY AND V. VEERAVALLI and β k α, ε = P k max Zk+n k 1 0 n<1 εl ε2 ql α α By condition 3.2, for every 0 < ε < 1 and a k 1, β k α, ε = P k max Zk+n k 1 εl α 0 n<1 εl α. 1 + εq 0. α 0 Next, for any τ α and n 1, α PFAτ P τ < n λ > n = P τ < n λ > n P λ > n = P τ < n Π n, and, therefore, 3.8 P τ < n α/π n, n 1. It foows that the first term in inequaity p k α, ε αe 1 ε2 ql α Π mαk 1, where m α k = k + 1 εl α is the greatest integer number k + 1 εl α. Since α = e q dl α and m α k k 1/1 ε L α m α k k/1 ε, we obtain p k α, ε Π 1 m exp αk d + ε2 q 1 ε m αk k 1 and og p k α, ε m α k By conditions 2.6 and 2.7, og Π m αk m α k d + ε2 q 1 ε m α k k 1. m α k og p k α, ε im d d + ε2 q α 0 m α k 1 ε = ε d + εq, 1 ε where d > 0 for π Ed and d = 0 for π H. It foows that, for any π H Ed, 3.10 p k α, ε 0 as α 0 for a k 1 and 0 < ε < 1. Therefore, we obtain that for every τ α and ε > 0, 3.11 γ ε,ατ k p k α, ε + β k α, ε, where by 3.7 and 3.10, β k α, ε and p k α, ε go to zero as α 0. Since both p k α, ε and β k α, ε do not depend on a particuar stopping time τ, 3.3 hods. Let N α = εl α be the greatest integer number εl α. Evidenty, 3.12 γ ε,α τ = Using 3.11 and 3.12, we obtain π k γ k ε,ατ N α π k γ k ε,ατ + Π Nα. N α 3.13 γ ε,α τ Π Nα + π k β k α, ε + sup p k α, ε. k N α The term Π Nα 0 as α 0. The second term goes to zero as α 0 by condition 3.2 see 3.7 and Lebesgue s dominated convergence theorem. It suffices to show that the third term vanishes as α 0. By 3.9, sup p k α, ε αe 1 ε2 ql α Π mα N α 1 = Π mα N α 1 exp d + ε 2 ql α, k N α where m α N α = N α + 1 εl α. Obviousy, L α m α N α L α + 2, and hence, og sup k Nα p k α, ε m α N α og Π m αn α m α N α L α d + ε 2 q L α + 2.

7 ASYMPTOTIC THEORY OF QUICKEST CHANGE DETECTION 7 Since we obtain that for any π H Ed, og Π m α N α m α N α d as α 0, og sup k Nα p k α, ε im ε 2 q, α 0 m α N α showing that sup k Nα p k α, ε 0 as α 0. Since the right side in 3.13 does not depend on τ, 3.4 foows, and the proof is compete. In the next theorem, we derive the ower bounds for the positive moments of the detection deay of any procedure from the cass α under the conditions postuated in Lemma 1. A simiar bound has been obtained by Lai [16] for E π τ λ + and heavy-taied prior distributions under a sighty stronger conditions with sup k in 3.2. Reca that q d = q + d when π Ed and q d = q when π H. THEOREM 1. Let condition 3.2 hod for some positive finite number q. Then, for every k 1 and m > 0, og α m 3.14 inf E k[τ k m τ k] 1 + o1 as α 0 τ α and for a m > inf τ α Eπ [τ λ m τ λ] where o1 0 as α 0. q d og α m 1 + o1 as α 0, PROOF. By the Chebyshev inequaity, for any 0 < ε < 1 and m > 0 Obviousy, Therefore, E k [τ k + ] m [1 εl α ] m P k τ k 1 εl α E k [τ k m τ k] = E k[τ k + ] m P k τ k 1 εl α = P k τ k γ k ε,ατ. P k τ k Next, by 3.8, for any τ α and α < Π k q d [1 εl α] m P k τ k 3.17 P k τ k = 1 P τ < k 1 α/π k. [ ] P k τ k γ ε,ατ k. Using 3.16 and 3.17, yieds the inequaity [ ] 3.18 E k [τ k m τ k] [1 εl α ] m 1 γ ε,ατ/1 k α/π k, which hods for any τ α, 0 < ε < 1, and k : Π k < α. By Lemma 1, γ k ε,ατ 0 as α 0 uniformy over a τ α, which impies inf E k[τ k m τ k] [1 εl α ] m 1 + o1 as α 0. τ α Since ε is arbitrary, the asymptotic ower bound 3.14 foows. To prove the asymptotic ower bound 3.15 we again use the Chebyshev inequaity, according to which for any 0 < ε < 1 and any τ α where E π [τ λ + ] m [1 εl α ] m P π τ λ 1 εl α, P π τ λ 1 εl α = P π τ λ γ ε,α τ.

8 8 A. TARTAKOVSKY AND V. VEERAVALLI Since P π τ λ 1 α for any τ α, it foows that 3.19 E π [τ λ m τ λ] = Eπ [τ λ + ] m P π τ λ [ [1 εl α ] m 1 γ ε,ατ 1 α [ [1 εl α ] m 1 γ ε,ατ P π τ λ Since ε can be arbitrariy sma and, by Lemma 1, sup τ α γ ε,α τ 0 as α 0, the asymptotic ower bound 3.15 foows, and the proof is compete. REMARK 1. It is important to emphasize that the vanishing term o1 in 3.14 depends on k. For this reason, the ower bound 3.15 does not foow directy from the inequaity 3.14, and an additiona effort was needed to prove it Asymptotic performance of the procedure ν B for arge B. We begin with the evauation of the performance of the Shiryaev detection procedure ν B for arge vaues of B regardess of the fase aarm constraint. Write Y t = og f 1,tX t X t 1 f 0,t X t X t 1, and, for every k = 1, 2,... and ε > 0, define the random variabe 3.20 T k ε ]. = sup n 1 : 1 k+n 1 Y t q n t=k > ε, where sup = 0. Ceary, in terms of T ε k, the amost sure convergence of 3.1 may be written as P k T ε k < = 1 for a ε > 0 and k 1, which impies condition 3.2. Whie these conditions are sufficient for obtaining ower bounds for moments of the detection deay in particuar, for the average detection deay, they need to be strengthened in order to estabish asymptotic optimaity properties of the detection procedure ν B, and to obtain asymptotic expansions for moments of the detection deay. Indeed, in genera these conditions do not even guarantee finiteness of CADD k ν B and ADDν B. In order to study asymptotics for the average detection deay one may impose the foowing constraints on the rate of convergence in the strong aw for Zk+n k /n: 3.21 E k T k ε < for a ε > 0 and k 1, and 3.22 Note that 3.21 is cosey reated to the condition n=1 k+n 1 P k t=k π k E k T k ε < for a ε > 0. Y t qn > εn < for a ε > 0 and k 1, which is nothing but the compete convergence of Z k k+n /n to q under P k cf. Hsu and Robbins [11]. We write this compacty as n Zk k+n P k competey q for every k 1. n The convergence condition 3.22 is a joint condition on the rates of convergence of Zk+n k /n for each λ = k and the prior distribution. We write this condition compacty as n Zλ P π average-competey λ+n q. n ]

9 ASYMPTOTIC THEORY OF QUICKEST CHANGE DETECTION 9 To study asymptotics for higher moments of the detection deay, the compete convergence conditions 3.23 and 3.24 shoud be further strengthened. A natura generaization is to require, for some r 1, 3.25 E k [T k ε ] r < for a ε > 0 and k 1 and 3.26 π k E k [T k ε ] r < for a ε > 0. If 3.25 hods, it is said that Zk+n k /n converges r quicky to q cf. Lai [13]. If 3.26 hods, we wi say that Zλ+n λ /n converges average-r-quicky to q. We wi write these modes of convergence compacty as 3.27 and n Zk P k r-quicky k+n q for every k 1 n 1 n Zλ P π average-r-quicky λ+n q. n Compete and r-quick convergence conditions have been previousy used by Lai [14], Tartakovsky [33], and Dragain, Tartakovsky and Veeravai [8] to estabish the asymptotic optimaity of sequentia hypothesis tests for genera statistica modes. Beow we take advantage of these resuts and prove that the conditions 3.23, 3.24, 3.27, and 3.28 are sufficient for asymptotic optimaity of the Shiryaev change-point detection procedure. In the foowing theorem, we estabish the operating characteristics of the detection procedure ν B for arge vaues of the threshod B regardess of the fase aarm rate constraints for genera statistica modes when the r-quick convergence conditions 3.27 and 3.28 hod. Hereafter X B Y B as B means that im B X B /Y B = 1. For the sake of compactness, in the rest of the paper we wi write ED π mτ for E π [τ λ m τ λ] and ED k m τ for E k [τ k m τ k], respectivey. THEOREM 2. i Let condition 3.27 hod for some positive q and r 1. Then for a m r og B m 3.29 ED k m ν B as B for a k 1. ii Let condition 3.28 hod for some positive q and r 1. Then for a m r og B m 3.30 ED π mν B as B. q d q d To prove this theorem we wi need two auxiiary resuts that we formuate in the form of Lemmas 2 and 3 beow. Lemma 2 is simiar to, and to some extent is a particuar case of, Lemma 1. Lemma 3 is reated to convergence of moments of one-sided stopping times that bound the stopping time ν B from above. The first emma wi be used to obtain ower bounds for the moments of the detection deay, whie the second one wi be used for deriving the corresponding upper bounds. The foowing notation is used throughout this subsection: L B = q 1 d og B, γ k ε B = P k k ν B < k + 1 εl B, γ ε B = P π λ ν B < λ + 1 εl B. Note that γ ε B = π k γ ε k B.

10 10 A. TARTAKOVSKY AND V. VEERAVALLI LEMMA 2. Suppose condition 3.2 hods for some q > 0. Then, im B γk ε B = 0 for a 0 < ε < 1 and k 1; im εb = 0 B for every 0 < ε < 1. PROOF. The proof runs aong the ines of the proof of Lemma 1. It suffices to note that P π ν B < λ 1/1 + B 1/B. Therefore, repacing α by 1/B in Lemma 1 competes the proof. We now formuate the second important resut that wi be used to obtain upper bounds for the moments of the stopping time ν B. Write S k k+n 1 = Zk k+n 1 + nw n,k, w n,k = n 1 ogπ k /Π k+n 1 and, for b > 0, introduce the sequence of one-sided stopping times 3.33 η b k = infn 1 : Sk+n 1 k b, k = 1, 2,... LEMMA 3. Let m be a positive, not necessariy integer number and suppose that for some r > 0 the condition 3.25 is satisfied. Then, for a m r, 3.34 E k ηb k/b m qd m as b. PROOF. For the geometric prior distribution with π 0 = 0, this emma can be directy derived from Theorem 4.2 of Dragain, Tartakovsky and Veeravai [8], since in this case og Π n /n = d = og1 ρ and w n,k = n og1 ρ for a n 1. In the genera case, the proof requires a modification which is given beow. By condition 3.25, Zk+n k /n q as n P k a.s. Since by assumptions , w n,k d as n d 0, it foows that Sk+n k /n q d as n P k a.s. for a k 1. Therefore, P k 1 M max 0 n<m Sk k+n 1 + εq d M 0 for a ε > 0 and k 1 and the argument identica to that used in the proof of Lemma 1 with α repaced by e b yieds Hence Chebyshev s inequaity for any m > 0 appies to show that P k η b k 1 εb/q d 1 as b. E k [η b k] m [1 εb/q d ] m P k η b k 1 εb/q d 3.35 E k [η b k] m b/q d m 1 + o1 as b. To obtain the upper bound, define = sup n 1 : n 1 Sk+n 1 k q d > ε. T k ε It is easy to see that S k k+η b k 2 < b and S k k+η b k 2 η bk 1q d ε on η b k 1 > k T ε. It foows that for every 0 < ε < q d η b k 1 + b q d ε 1 k k + T η b k> T ε +1 ε + 11 k ηb k T ε k T ε b q d ε. Since w n,k d as n, the condition 3.25 impies E k [ etting ε 0 we obtain that for a m r E k [η b k] m b/q d m 1 + o1 as b, which, aong with the reverse inequaity 3.35, competes the proof. k T ε ] r <. Since ε can be arbitrariy sma,

11 We are now ready to prove Theorem 2. ASYMPTOTIC THEORY OF QUICKEST CHANGE DETECTION 11 PROOF OF THEOREM 2. i For b = og B, define the sequence of stopping times η b k as in It is easiy verified that the statistic og Λ n can be written in the form 3.37 og Λ n = Z k n + n k + 1w k+n 1,k + n,k, where the random variabe n,k is nonnegative. Thus, for b = og B, 3.38 ν B k η b k on ν B k for every k 1 and 3.39 E k [ν B k + ] m E k [η b k] m ; ν B k E k [η b k] m for every k 1. By 2.4 and 3.8, P k ν B k 1 [1 + BΠ k ] 1, and hence, for B > 1 Π k /Π k, ED k m ν B = E k[ν B k + ] m P k ν B k E k [η b k] m 1 [1 + BΠ k ] 1. Appying Lemma 3, we obtain the foowing estimate for the upper bound: og B m 3.40 ED k m ν B 1 + o1 as B for a k 1. q d In order to prove 3.29 it remains to show that the right side of the atter inequaity is a ower bound for ED k m ν B as B. To that end, we use Lemma 2 and the Chebyshev inequaity. Indeed, appying the Chebyshev inequaity and the inequaity P k ν B k 1 [1 + BΠ k ] 1 yieds, for B > 1 Π k /Π k, [ ] ED k m ν B [1 εl B ] m P k ν B k γ ε k B [ ] [1 εl B ] m 1 1/Π k B γ ε k B. Since ε is arbitrary and, by Lemma 2, γ ε k B 0 as B for every k 1, we obtain that the right side in 3.40 is the asymptotic ower bound for ED k m ν B. The asymptotic formua 3.29 foows. ii Simiar to 3.36, for every 0 < ε < q d and k 1, 3.41 ν B k η b k T k ε b q d ε on ν B k. Appying 3.41 aong with the fact that P π ν B λ B/1 + B yieds ED π mν B B + 1 og B m k π k E k + T ε + 2. B q d ε Since ε can be arbitrariy sma and, by the assumption of the theorem, π ke k [T ε k ] m <, which as before impies π k ke k [ T ε ] m <, it foows that og B m 3.42 ED π mν B 1 + o1 as B. q d To obtain a ower bound for ED π mν B, we again use the Chebyshev inequaity, which yieds ED π mν B E π [ν B λ + ] m [1 εl B ] m [P π ν B λ γ ε B] [1 εl B ] m [B/1 + B γ ε B], where γ ε B 0 as B by Lemma 2. Reca that L B = og B/q d. Since ε is arbitrary, it foows that og B m 3.43 ED π mν B 1 + o1 as B, which aong with the upper bound 3.42 proves The proof is compete. q d

12 12 A. TARTAKOVSKY AND V. VEERAVALLI REMARK 2. The proof suggests that 3.29 hods for a k < K α where K α is such integer number for which Pλ > K α < α. From a theoretica viewpoint this does not cause the probem, since K α as α 0. However, from the point of view of practica appications it is important to investigate the behavior of ED k m ν B for arge vaues of k Asymptotic optimaity. We are now in a position to prove the asymptotic optimaity of the detection procedure ν B with the threshod B = 1 α/α in the cass α for sma vaues of the PFA α. Since everything we need has been prepared, the proof is immediate. THEOREM 3. If B = B α = 1 α/α, then the detection procedure ν Bα beongs to the cass α, and the foowing two assertions hod. i Let condition 3.27 hod for some positive q and r 1. Then for a m r and k 1, 3.44 inf τ α EDk m τ ED k m ν Bα og α m as α 0. ii Let condition 3.28 hod for some positive q and r 1. Then for a m r, og α m 3.45 inf τ α EDπ mτ ED π mν Bα as α 0. PROOF. Appying Theorems 1 and 2 yieds 3.44 and COROLLARY 1. Let B = B α = 1 α/α. If condition 3.23 is satisfied for some positive q, then 3.46 inf CADD og α kτ CADD k ν Bα as α 0 for a k 1. τ α q d If condition 3.24 is satisfied for some positive q, then 3.47 inf ADDτ ADDν og α B α as α 0. τ α q d We stress that the detection procedure ν B with the threshod B = B α = 1 α/α is asymptoticay optima not ony reative to the ADD, but aso uniformy asymptoticay optima with respect to the conditiona ADD for a vaues of λ = k, k = 1, 2,.... We obtain this strong optimaity resut primariy because the constraint on fase aarms in the Bayesian formuation that we consider is averaged over a possibe reaizations of the change point. Such a strong optimaity resut is not avaiabe for the minimax formuation of the probem with the constraint on the mean time to fase aarm E τ [16, 17, 21, 34]. It is aso interesting to observe from Theorem 3 that for prior distributions with exponentia tai d > 0 if q d, the observations contain more information about the change than the prior distribution, and the performance is determined by q. On the other hand, if q d, then the decision about the change-point can be made based soey on the prior distribution to yied an ADD of og α /d. For heavy-taied distributions π H, prior information does not affect asymptotic performance, as coud be expected. REMARK 3. The resuts of Theorem 3 remain true in a more genera case where the r-quick convergence condition 3.27 is satisfied with an increasing function φn in pace of n, i.e φn Zk P k r-quicky k+n q for a k 1. n In particuar, if 3.48 hods with φn = n p, p > 0, then, simiar to 3.44, the foowing more genera resut can be proved: og α m/p inf τ α EDk m τ ED k m ν Bα as α 0. It is worth noting that the r-quick convergence conditions 3.27 and 3.28 are ony sufficient and by no means are necessary. Indeed, the proof of Lemma 3 suggests that the upper bound hods whenever eft-sided versions of r-quick conditions are satisfied, i.e. E k [t k ε ] r < and π ke k [t k ε ] r < where = sup n 1 : n 1 Zk+n 1 k q < ε. t k ε q d q d q d

13 ASYMPTOTIC THEORY OF QUICKEST CHANGE DETECTION 13 Moreover, even these atter conditions can be substantiay reaxed into the foowing condition: 3.49 im n nr 1 P k n 1 Zk+n 1 k q ε = 0 for a ε > 0, k 1, and some r 1. A proof can be buit upon a generaization of a trick expoited recenty by Lai [16] for studying the asymptotic optimaity of the CUSUM and window-imited CUSUM tests. However, we find it natura and convenient from the methodoogica standpoint to formuate conditions in terms of rates of convergence in the strong aw of arge numbers for the og-ikeihood ratio process. Besides, r-quick convergence impies the right-tai condition 3.2 which is the key for obtaining the ower bounds. 4. Asymptotic Operating Characteristics of the Detection Procedure ν B in the i.i.d. Case In this section, we wi dea with the i.i.d. case where f 0,n X n X n 1 = f 0 X n and f 1,n X n X n 1 = f 1 X n. Then P is the probabiity measure under which the pdf of X n is f 0 x for every n 1 and for λ = k, k 1, P k is the probabiity measure under which the pdf of X n is f 0 x if n k 1 and is f 1 x if n k with respect to a σ-finite measure µx. The LLR defined in 2.5 gets modified to 4.1 Z k n := og dp k dp X n = n t=k og f 1X t, n k, f 0 X t and the decision statistic Λ n obeys 2.9 with Zn k defined in 4.1. Note aso that in the i.i.d. case the statistic Λ n satisfies the recursion 4.2 Λ n = Πn 1 Π n Λ n 1 + π n Π n f1 X n f 0 X n, n 1, Λ 0 = π 0 /1 π 0, which may be depoyed for practica impementation and simuations. As it was mentioned in Section 2, in the i.i.d. case and for the geometric prior distribution, the Shiryaev procedure 2.10 is optima when the threshod B can be chosen in such a way that PFAν B = α. Since it is difficut to meet this exact requirement, we wi study the properties of the detection procedure ν B with B = 1 α/α, which guarantees the inequaity PFAν B α. Since this choice negects the overshoot, it is expected that the actua fase aarm probabiity may be substantiay smaer than α see Sections 4.2 and 6. Let Df 1, f 0 = E 1 Z1 1 f1 x = og f 1 xµdx f 0 x be the Kuback-Leiber K-L information number between the densities f 1 x and f 0 x aso caed the K-L distance. In the i.i.d. case, the K-L number Df 1, f 0 pays the roe of the number q that appeared in Theorems 1-3 of the previous section. By anaogy with 3.20 we define the ast entry times 4.3 T k ε = sup n 1 : 1 k+n 1 Y t Df 1, f 0 > ε n t=k Since Y t, 1, 2,... are i.i.d. random variabes, T ε k have the same statistica properties for a k 1. Therefore, E k [T ε k ] r = E 1 [T ε 1 ] r and the condition E 1 [T ε 1 ] r < for a ε > 0 i.e. the r quick convergence condition of Zn/n 1 to Df 1, f 0 under P 1 impies 4.4. π k E k [T k ε ] r = 1 π 0 E 1 [T 1 ε ] r < for a ε > 0. In the i.i.d. case, the condition E 1 Z 1 1 r+1 < is both necessary and sufficient for the r quick convergence of Z 1 n/n to Df 1, f 0. Indeed, by the Baum-Katz rate of convergence in the strong aw [2], the foowing

14 14 A. TARTAKOVSKY AND V. VEERAVALLI statements are equivaent for any r > 0: 4.5 E 1 Z 1 1 r+1 < n r 1 n P 1 Ỹ t εn < for some ε > 0, n=1 t=1 n r 1 sup P 1 t 1 Ỹ t ε < for a ε > 0, n=1 t n where Ỹt = Y t Df 1, f 0. Since P 1 T ε 1 > n P 1 sup t n t 1 Ỹ t ε, the impication E 1 Z1 1 r+1 < E 1 [T ε 1 ] r < for a ε > 0 foows. Appying Theorem 3, one can concude that if the K-L number Df 1, f 0 is stricty positive and the r+1 st absoute moment of the LLR Z1 1 is finite, then the Shiryaev detection procedure ν B α asymptoticay minimizes moments of the detection deay up to the order r. However, beow we show that the Shiryaev procedure minimizes a positive moments of the detection deay under weaker conditions. As we demonstrate, a that is required is positiveness and finiteness of the K-L numbers. Detais are given in the next subsection Asymptotic optimaity. We wi impose the foowing mid condition on the K-L information numbers: < Df 1, f 0 < and 0 < Df 0, f 1 <. The first positiveness of the K-L information numbers is not at a restrictive, since it hods whenever the pdfs f 0 x and f 1 x do not coincide amost everywhere, i.e. µ x : f 0 x f 1 x > 0. If it does not hod, the LLR Z1 1 is equa to zero amost surey, in which case the detection probem is degenerate. The second condition finiteness is quite natura and it hods in most cases. However, there are reasonabe modes for which it does not hod. The probem of detecting a change in mean vaue of a rectanguar distribution may serve as a good exampe. In the atter case, the moments of the LLR are infinite and the detection probem becomes degenerate at east in the asymptotic setting. On the other hand, if K-L numbers are finite, then a the moments of the negative part of the LLR are finite, E 1 min0, Z1 1m <, since the pdfs f 0 x and f 1 x are mutuay absoutey continuous i.e. if f 0 x = 0 so is f 1 x, which impies that f 0 X E f1 f 1 X 1 = E 1 exp Z1 1 = 1. In the rest of this section, for the sake of simpicity, we wi restrict our attention to the geometric prior distribution given in 2.8, in which case d = og1 ρ and the statistic Λ n obeys the recursion 4.8 Λ n = 1 1 ρ Λ n 1 + ρ f 1X n f 0 X n, n 1, Λ 0 = π 0 /1 π 0. The atter recursion foows from 4.2 observing that π n /Π n = ρ/1 ρ and Π n 1 /Π n = 1/1 ρ. The foowing theorem estabishes asymptotic optimaity properties of ν Bα in the cass α with respect to a positive moments of the detection deay. THEOREM 4. Let, conditioned on λ = k, the observations X 1, X 2,..., X k 1 be i.i.d. with the pdf f 0 x and X k, X k+1 be i.i.d. with the pdf f 1 x. Further, et prior distribution of the change point λ be geometric. Suppose that conditions 4.6 are satisfied. If B α = 1 α/α, then PFAν Bα α and, as α 0, for a m inf τ α EDπ mτ ED π mν Bα og α Df 1, f 0 + og1 ρ 4.10 inf τ α EDk m τ ED k og α m m ν Bα k 1. Df 1, f 0 + og1 ρ m,

15 ASYMPTOTIC THEORY OF QUICKEST CHANGE DETECTION 15 PROOF. For b > 0, define η b k as in 3.33 and note that in the i.i.d. case and for the geometric prior distribution the statistic Sk+n 1 k ρ = Zk k+n 1 + n og1 ρ, n 1 is a random wak with mean E ksk kρ = Df 1, f 0 + og1 ρ. By conditions 4.6, Df 1, f 0 is positive and finite and hence E k min0, Zk km < for a m > 0. Indeed, E k exp min0, Zk k = E k e Zk k 1Z k k <0 + E k1 Z k k 0 E ke Zk k + 1 = 2, where the ast equaity foows from 4.7. Therefore, Theorem III.8.1 of Gut [10] appies to show that for a m 1 E k [η b k] m b m = 1 + o1 as b. Df 1, f 0 + og1 ρ Since ν B k η og B k on ν B k and P k ν B k 1 as B, it foows that ED k og α m m ν Bα 1 + o1 as α 0. Df 1, f 0 + og1 ρ Note that in the i.i.d. case condition 3.2 hods triviay with q = Df 1, f 0, since Zk+n k /n converges to Df 1, f 0 P k a.s. by the strong aw of arge numbers and since P k 1 M max 0 n<m Zk k+n 1 + εdf 1, f 0 does not depend on k. Thus, the ower bound foows from 3.14: og α m inf τ α EDk m τ 1 + o1 as α 0, Df 1, f 0 + og1 ρ which competes the proof of The proof of 4.9 is quite simiar and is therefore omitted Higher-order asymptotic approximations for ADD and PFA. In this subsection, we use the noninear renewa theory deveoped by Woodroofe [42] see aso Siegmund [29] to improve the first-order approximations for the ADD and PFA. We wi aso suppose with minor oss of generaity that π 0 = 0, i.e. in the rest of this subsection the pure geometric prior distribution, π k = ρ1 ρ k 1, k 1, wi be considered. We first observe that, in this case, CADD 1 ν B CADD k ν B for a k 1. To understand why, it is sufficient to consider the recursion 4.8 and note that, for λ = k = 1, the initia condition Λ 0 = 0 whie, for λ = k 2, 0 Λ k 1 < B on ν B k. Moreover, for arge B, the difference between E 1 ν B 1 and E k ν B k ν B k is a constant that is approximatey equa to the mean of the initia condition, E og Λ k 1 for k = 1 this vaue is equa to 0. This constant varies for different modes and its cacuation is usuay probematic. For this reason, we wi concentrate on the evauation of the worst-case deay CADD 1 ν B. In order to appy reevant resuts from noninear renewa theory, we rewrite the stopping time ν B in the form of a random wak crossing a constant threshod pus a noninear term that is sowy changing in the sense defined by Woodroofe [42] and Siegmund [29]. Indeed, the stopping time ν B can be written in the foowing form 4.11 ν B = inf n 1 : S n ρ + n b, b = ogb/ρ, where S n ρ = Z n + n og1 ρ is a random wak with mean E 1 S 1 ρ = Df 1, f 0 + og1 ρ and n 1 i n n = og ρ i f 0 X s = og ρ i e Z i. f 1 X s i=1 s=1 Here and in the rest of this subsection, we write Z n in pace of Z 1 n. For b > 0, define η b 1 = η b as in 3.33, i.e η b = infn 1 : S n ρ b, i=1

16 16 A. TARTAKOVSKY AND V. VEERAVALLI and et κ b = S ηb ρ b on η b < denote the excess overshoot of the statistic S n ρ over the threshod b at time n = η b. Let 4.14 Gy, ρ, D = im P 1 κ b y b be the imiting distribution of the overshoot and et κρ, D = im b E 1 κ b = denote the reated imiting average overshoot. Let us aso define and ζρ, D = im b E 1 e κ b = 4.15 Cρ, D = E 1 og y dgy, ρ, D e y dgy, ρ, D 1 ρ i e Z i. Note that by 4.11, S νb ρ = b νb + χ b on ν B <, where χ b = S νb ρ + νb b is the excess of the process S n ρ + n over the eve b at time ν B. Taking the expectations on both sides and appying Wad s identity, we obtain 4.16 D + og1 ρ E 1 ν B = b E 1 νb + E 1 χ b. The crucia observations are that the sequence n, n 1 is sowy changing, and that n converges P 1 -a.s. as n to the random variabe = og ρ i e Z i with finite expectation E 1 = Cρ, D. In fact, appying Jensen s inequaity yieds 4.17 Cρ, D = E 1 og ρ k = og1/ρ. i=1 Moreover, im n E 1 n = Cρ, D due to the fact that n. An important consequence of the sowy changing property is that, under mid conditions, the imiting distribution of the excess of a random wak over a fixed threshod does not change by the addition of a sowy changing noninear term see Theorem 4.1 of Woodroofe [42]. Furthermore, since n and E 1 n Cρ, D, using 4.16 we expect that for arge b, 1 E 1 ν B b Cρ, D + κρ, D. Df 1, f 0 + og1 ρ The mathematica detais are given in Theorem 5 beow. More importanty, noninear renewa theory aows us to obtain an asymptoticay accurate approximation for PFAν B that takes the overshoot into account. This approximation is important for practica appications where the vaue of Df 1, f 0 is moderate. For sma vaues of ρ and Df 1, f 0 the overshoot can be negected, and formua 2.11 wi be reasonaby accurate. THEOREM 5. Let the prior distribution of the change point λ be geometric, π k = ρ1 ρ k 1, k 1, and assume that Z n, n 1 are nonarithmetic with respect to P and P 1. i If conditions 4.6 hod, then ζρ, D 4.18 PFAν B = 1 + o1 as B. B ii If, in addition, the second moment E 1 Z 1 2 is finite, then as B 1 [ ] 4.19 E 1 ν B = ogb/ρ Cρ, D + κρ, D + o1. Df 1, f 0 + og1 ρ i=1

17 PROOF. i Obviousy, ASYMPTOTIC THEORY OF QUICKEST CHANGE DETECTION 17 PFAν B = E π 1 p νb = E π 1 + Λ νb 1 = E π [1 + BΛ νb /B] 1 = 1 B Eπ e χ b 1 + o1 as B, where χ b = og Λ νb b. Since χ b 0 and PFAν B 1/1 + B < 1/B, it foows that E π e χ b = E π e χ b ν B < λ PFAν B + E π e χ b ν B λ 1 PFAν B = E π e χ b ν B λ + O1/B as B. Therefore, it suffices to evauate the vaue of E π e χ b ν B λ = Pλ = k ν B ke k e χ b ν B k. To this end, we reca that, by 3.37, for any 1 k <, ν B = n 1 : Snρ k + n,k b, where S k nρ = Z k n + n k + 1 og1 ρ, n k is a random wak with the expectation E k S k k ρ = Df 1, f 0 + og1 ρ and n,k, n k are sowy changing under P k. Since, by conditions 4.6, 0 < Df 1, f 0 <, we can appy Theorem 4.1 of Woodroofe [42] to obtain im B E k e χ b ν B k = e y dgy, ρ, D = ζρ, D. 0 Aso, im Pλ = k ν π k P ν B k B k = im B B P π = π k. ν B λ Consequenty, im B Eπ e χ b ν B λ = im B Eπ e χ b = ζρ, D, which competes the proof of ii The proof of 4.19 rests on Woodroofe s Noninear Renewa Theorem see Theorem 4.5 in [42]. Indeed, by 4.11, the stopping time ν B is based on the threshoding of the sum of the random wak S n ρ and the noninear term n. Since P 1 -a.s. n and E 1 n E 1 = Cρ, D, n n n, n 1 are sowy changing under P 1. In order to appy this theorem we have to check the vaidity of the foowing three conditions: n=1 P 1 n εn < for some 0 < ε < D; max n+k, n 1, are P 1 -uniformy integrabe; 0 k n im b P 1 ν B εb/d + µ = 0 for some 0 < ε < 1. B Condition 4.20 hods triviay, since n 0. Since n, n = 1, 2... are non-decreasing, max 0 k n n+k = 2n and to prove 4.21 it suffices to show that n, n 1 are P 1 -uniformy integrabe. Since n and, by 4.17, E 1 <, the desired uniform integrabiity foows. Therefore, condition 4.21 is satisfied. We now turn to checking condition Noting that in the notation of Subsection 3.2, P 1 ν B < 1 εb/d ρ = γ ε 1 B, where D ρ = D + og1 ρ, and using inequaities 3.6 and 3.9 with α = e b, we obtain P 1 ν B < 1 εb/d ρ e y εb + β 1 ε, B,

18 18 A. TARTAKOVSKY AND V. VEERAVALLI where y ε > 0 for a ε > 0 and β 1 ε, B = P 1 max Z n 1 + εdk ε,b 1 n<k ε,b, K ε,b = 1 εb/d ρ. The first term in the above inequaity is o1/b as B. A it remains to do is to show that the second term is o1/b. To this end, we appy Theorem 1 of Chow and Lai [6], according to which for a ε > 0 and r 0 n r 1 P 1 max Z [ k Dk εn C r E 1 Z1 D +] r+1 [ + E1 Z 1 D 2] r 1 k n n=1 where C r is a universa constant. Reca that by the conditions of the theorem, E 1 Z 1 2 <. Therefore, the sum on the eft side of the previous inequaity is finite for r = 1 and a ε > 0, which impies that the summand shoud be o1/n. Since β 1 ε, B P 1 max Z n Dn εdf 1, f 0 K ε,b, n<k ε,b it foows that β 1 ε, B = ob 1. Hence condition 4.22 hods for a 0 < ε < 1. Thus, a the conditions of Theorem 4.5 in [42] are satisfied. The use of this theorem yieds 4.19 for arge B. REMARK 4. The constants κρ, D and ζρ, D are the subject of the renewa theory. The constant Cρ, D is not easy to compute in genera. For ρ cose to 1, the upper bound 4.17 may be usefu. Obviousy, this bound is asymptoticay accurate when D 0. Monte Caro experiments may be used to estimate C with a reasonabe accuracy see Tabe 1 in Section 6. The usefuness of Theorem 5 is two-fod. First, it provides accurate approximations for both the ADD and the PFA. Second, it aows us to study the important imiting case of ρ 0. To anayze the atter case it is convenient to consider the statistic R ρ,n = Λ n /ρ. As ρ 0 this statistic converges to the so-caed Shiryaev-Roberts statistic R n = im ρ 0 R ρ,n. Aso, as ρ 0, PFA tends to 1 and im ρ 0 [1 PFAν B ]/ρ = E ν B. Thus, it is natura to consider asymptotics α 1 and ρ 0 so that the ratio 1 α/ρ remains constant at T, 0 < T <, where T may be interpreted as a constraint on the mean time to fase aarm E ν B. The requirement 1 α/ρ T as α 1, ρ 0 is simiar to that used by Shiryaev [25] for detecting changes in stationary regimes for the Wiener process. The foowing coroary foows directy from Theorem 5 and the above argument. COROLLARY 2. Let ν B = infn : R ρ,n B and B = 1 α/αρ. Assume that α 1 and ρ 0 in such a way that 1 α/ρ T, where 0 < T <. Then, under the conditions of Theorem 5 1 [ ] 4.23 E 1 ν T = og T CD + κd + o1 as T, Df 1, f 0 where ν T represents ν B under the above imit, and where CD = C0, D and κd = κ0, D. It can be aso shown that in the conditions of Theorem 5, E ν B T as B = T ζd and T where ζd = ζ0, D. These resuts generaize simiar resuts obtained previousy by Poak [22] for exponentia famiies see Theorem 3 in [22]. 5. Asymptotic Performance of Other Detection Procedures It is known that in the case where the observations are i.i.d. and the change point is modeed as deterministic but unknown, the cumuative sum CUSUM detection procedure of Page [19] and the randomized Shiryaev- Roberts detection procedure proposed by Poak [21] are optima with respect to the minimax expected detection ag, subject to a constraint on the mean time to fase aarm. More specificay, consider the foowing two detection procedures ˆτ B = inf n 0 : R n B and τ B = inf n 1 : U n B,

19 ASYMPTOTIC THEORY OF QUICKEST CHANGE DETECTION 19 where the statistics R n and U n are defined as foows n R n = e Zk n and U n = max n. 1 k n ezk If R 0 = 0, the detection procedure ˆτ B is the Shiryaev-Roberts procedure. Its randomized version, when R 0 is random with a certain distribution, has been suggested by Poak [21]. We wi refer to this procedure to as the Shiryaev-Roberts-Poak SRP test. The second test τb is nothing but the CUSUM agorithm. In 1971, Lorden [17] proposed to measure the oss due to the detection deay by the worst-worst minimax risk ESτ = sup k ess sup E k τ k X k 1 and showed that the CUSUM procedure τb with B = T is asymptoticay optima for i.i.d. modes, as T, in the cass T = τ : E τ T of procedures for which the mean time to fase aarm E τ exceeds a predefined number T. In 1986, Moustakides [18] proved that the CUSUM test is stricty optima for any T > 0 with respect to the risk ESτ whenever the threshod B is chosen so that E τb = T. In 1996, Shiryaev [28] extended this resut for continuous-time processes detecting a change in the mean of the Brownian motion. In 1998, Lai [16] generaized previous resuts for a genera non-i.i.d. case showing that the CUSUM procedure is asymptoticay optima in the cass T as T with respect to the worst-worst risk ESτ as we as with respect to the average-worst risk minimax deay MDτ = sup k E k τ k τ k. The atter measure of the detection speed has been introduced earier by Poak [21]. In the mid of 1980s, Poak [21] proposed the randomized version of the Shiryaev-Roberts procedure ˆτ B where the statistic R n is randomized at the zero point n = 0, i.e. R 0 is a random variabe R 0 = 0 for the standard Shiryaev-Roberts procedure. Poak proved that this randomized procedure is neary optima with respect to the risk MDτ for i.i.d. data modes. Poak [22] aso presented a comprehensive asymptotic anaysis of the procedure ˆτ B for exponentia famiies. See [1],[30]-[39] for further extensions and detais. Therefore, both the CUSUM and the SRP detection procedures minimize sup k E k τ k τ k the expected detection deay in the worst-case scenario in the cass of procedures for which the mean time to fase aarm E τ exceeds a predefined number T at east as T. In this section, we study asymptotic properties of these two cassica change-point detection procedures in the cass α. The resuts of previous sections are used to estabish that they ose the asymptotic optimaity property under the Bayesian criterion for prior distributions with exponentia tai Ed, but remain optima for heavy-taied prior distributions H. In order to obtain an upper bound for PFA of the SRP procedure, we note that the statistic R n n is a zeromean P -martingae with respect to Fn X. Therefore, R n is a submartingae with mean E R n = n. Using Doob submartingae inequaity, we get P ˆτ B < n = P max R k B n/b, 1 k<n which yieds PFAB = P π ˆτ B < λ = π n P ˆτ B < n λ/b, n=1 where λ = kπ k is the mean of the prior distribution. Thus, choosing B = B α = λ/α guarantees ˆτ Bα α. Since τ B ˆτ B, for the CUSUM procedure we obtain 5.1 P τ B < n P ˆτ B < n n/b, n 1, and hence, 5.2 PFA B = P π τ B < λ = π n P τb < n B 1 n=1 n=1 nπ n = λ/b. It foows that B = B α = λ/α impies τ Bα α. The foowing theorem, which is a prototype of Theorem 3, estabishes the asymptotic operating characteristics of the CUSUM and SRP procedures in terms of moments of the detection deay.