ASYMPTOTICS OF MARKOV KERNELS AND THE TAIL CHAIN

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1 ASYMPTOTICS OF MARKOV KERNELS AND THE TAIL CHAIN SIDNEY I. RESNICK AND DAVID ZEBER Abstract. An asyptotic odel for extree behavior of certain Markov chains is the tail chain. Generally taking the for of a ultiplicative rando walk, it is useful in deriving extreal characteristics such as point process liits. We place this odel in a ore general context, forulated in ters of extree value theory for transition kernels, and extend it by foralizing the distinction between extree and non-extree states. We ake the link between the update function and transition kernel fors considered in previous work, and we show that the tail chain odel leads to a ultivariate regular variation property of the finite-diensional distributions under assuptions on the arginal tails alone. 1. Introduction A ethod of approxiating the extreal behavior of discrete-tie Markov chains is to use an asyptotic process called the tail chain under an asyptotic assuption on the transition kernel of the chain. Loosely speaking, if the distribution of the next state converges under soe noralization as the current state becoes extree, then the Markov chain behaves approxiately as a ultiplicative rando walk upon leaving a large initial state. This approach leads to intuitive extreal odels in such cases as autoregressive processes with rando coefficients, which include a class of ARCH odels. The focus on Markov kernels was introduced by Sith 24]. Perfekt 18, 19] extended the approach to higher diensions, and Segers 23] rephrased the conditions in ters of update functions. Though not restrictive in practice, the previous approach tends to ask aspects of the processes extreal behaviour. Markov chains which adit the tail chain approxiation fall into one of two categories. Starting fro an extree state, the chain either reains extree over any finite tie horizon, or will drop to a non-extree state of lower order after a finite aount of tie. The latter case is probleatic in that the tail chain odel is not sensitive to possible subsequent jups fro a non-extree state to an extree one. Previous developents handle this by ruling out the class of processes exhibiting this behaviour via a technical condition, which we refer to as the regularity condition. Also, ost previous work has assued stationarity, since interest focused on coputing the extreal index or deriving liits for the exceedance point processes, drawing on the theory established for stationary processes with ixing by Leadbetter et al. 17]. However, stationarity is not fundaental in deterining the extreal behaviour of the finite-diensional distributions. We place the tail chain approxiation in the context of an extree value theory for Markovian transition kernels, which a priori does not necessitate any such restrictions on the class of processes to which it ay be applied. In particular, we introduce the concept of boundary distribution, which controls tail chain transitions fro non-extree to extree. Although distributional convergence results are ore naturally phrased in ters of transition kernels, we treat the equivalent update function fors as an integral coponent to interfacing with applications, and we phrase relevant Key words and phrases. Extree values, ultivariate regular variation, Markov chain, transition kernel, tail chain, heavy tails. S. I. Resnick and D. Zeber were partially supported by ARO Contract W911NF and NSA Grant H at Cornell University. 1

2 2 S. I. RESNICK AND D. ZEBER assuptions in ters of both. While not aking explicit a coplete tail chain odel for the class of chains excluded previously, we deonstrate the extent to which previous odels ay be viewed as a partial approxiation within our fraework. This is accoplished by foralizing the division between extree and non-extree states as a level we ter the extreal boundary. We show that, in general, the tail chain approxiates the extreal coponent, the portion of the original chain having yet to cross below this boundary. Phrased in these ters, the regularity condition requires that the distinction between the original chain and its extreal coponent disappears asyptotically. After introducing our extree value theory for transition kernels, along with a representation in ters of update functions, we derive liits of finite-diensional distributions conditional on the initial state, as it becoes extree. We then exaine the effect of the regularity condition on these results. Finally, adding the assuption of arginal regularly varying tails leads to convergence results for the unconditional distributions akin to regular variation Notation and Conventions. We review notation and relevant concepts. If not explicitly specified, assue that any space S under discussion is a topological space paired with its Borel σ-field of open sets BS to for a easurable space. Denote by KS the collection of its copact sets; by CS the space of real-valued continuous, bounded functions on S; and by C + K S the space of non-negative continuous functions with copact support. Weak convergence of probability easures is represented by. For a space E which is locally copact with countable base for exaple, a subset of, ] d, M + E is the space of non-negative Radon easures on BE; point easures consisting of single point asses at x will be written as ɛ x. A sequence of easures {µ n } M + E converges v vaguely to µ M + E written µ n µ if E f dµ n E f dµ as n for any f C+ K E. The shorthand µf = f dµ is handy. That the distribution of a rando vector X is regularly varying on a cone E, ] d v \{0} eans that t PX/bt ] µ in M + E as t for soe non-degenerate liit easure µ M + E and scaling function bt. The liit µ is necessarily hoogeneous in the sense that µ c = c α µ for soe α > 0. The regular variation is standard if bt = t. If X = X 0, X 1, X 2,... is a hoogeneous Markov chain and K is a Markov transition kernel, we write X K to ean that the dependence structure of X is specified by K, i.e. PX n+1 X n = x] = K x,, n = 0, 1,.... We adopt the standard shorthand P x X 1,..., X ] = PX 1,..., X X 0 = x]. Soe useful technical results are assebled in Section 8 p Extreal Theory for Markov Kernels We begin by focusing on the Markov transition kernels rather than the stochastic processes they deterine, and introduce a class of kernels we ter tail kernels, which we will view as scaling liits of certain kernels. Antecedents include Segers 23] definition of back-and-forth tail chains that approxiate certain Markov chains started fro an extree value. For a Markov chain X K on 0,, it is reasonable to expect that extreal behaviour of X is deterined by pairs X n, X n+1, and one way to control such pairs is to assue that X n, X n+1 belongs to a bivariate doain of attraction cf. 5, 24]. In the context of regular variation, writing 2.1 t P Xn bt A 0, X ] n+1 bt A 1 = K btu, bta 1 t P A 0 Xn bt du ]

3 ASYMPTOTICS OF MARKOV KERNELS AND THE TAIL CHAIN 3 suggests cobining arginal regular variation of X n with a scaling kernel liit to derive extreal properties of the finite-diensional distributions fdds 18, 19, 23], and this is the direction we take. We first discuss the kernel scaling operation. For siplicity, we assue the state space of the Markov chain is 0,, although with suitable odifications, it is relatively straightforward to extend the results to R d. Henceforth G and H will denote probability distributions on 0, Tail Kernels. The tail kernel associated with G, with boundary distribution H, is 2.2 K y, A { Gy 1 A y > 0 = HA y = 0 for any easurable set A. Thus, the class of tail kernels on 0, is paraeterized by the pair of probability distributions G, H. Such kernels are characterized by a scaling property: Proposition 2.1. A Markov transition kernel K is a tail kernel associated with soe G, H if and only if it satisfies the relation 2.3 K uy, A = K y, u 1 A when y > 0 for any u > 0, in which case G = K1,. The property 2.3 extends to y = 0 iff H = ɛ 0. Proof. If K is a tail kernel, 2.3 follows directly fro the definition. Conversely, assuing 2.3, for y > 0 we can write K y, A = K 1, y 1 A, deonstrating that K is a tail kernel associated with K1, with boundary distribution H = K0,. To verify the second assertion, fixing u > 0, we ust show that Hu 1 = H iff H = ɛ 0. On the one hand, we have ɛ 0 u 1 A = ɛ 0 A. On the other, H0, = li n Hn 1, = H1,, so H0, 1] = 0. A siilar arguent shows that H1, = 0 as well. We call the Markov chain T K a tail chain associated with G, H. Such a chain can be represented as 2.4 T n = ξ n T n 1 + ξ n 1 {Tn 1 =0} for n = 1, 2,..., where ξ n iid G and ξ n iid H are independent of each other and of T0. If H = ɛ 0, then T becoes a ultiplicative rando walk with step distribution G and absorbing barrier at {0}: T n = T 0 ξ 1 ξ n Convergence to Tail Kernels. The tail chain approxiates the behaviour of a Markov chain X K in extree states. Asyptotic results require that the noralized distribution of X 1 be well-approxiated by soe distribution G when X 0 is large, and we interpret this requireent as a doain of attraction condition for kernels. Definition. A Markov transition kernel K : 0, B0, 0, 1] is in the doain of attraction of G, written K DG, if as t, 2.5 K t, t G on 0, ]. Note that DG contains at least the class of tail kernels associated with G i.e. with any boundary distribution H. A siple scaling arguent extends 2.5 to 2.6 K tu, t Gu 1 =: K u,, u > 0, where K is any tail kernel associated with G; this is the for appearing in 2.1. Thus tail kernels are scaling liits for kernels in a doain of attraction. In fact, tail kernels are the only possible liits:

4 4 S. I. RESNICK AND D. ZEBER Proposition 2.2. Let K be a transition kernel and H be an arbitrary distribution on 0,. If for each u > 0 there exists a distribution G u such that K tu, t G u as t, then the function K defined on 0, B0, as is a tail kernel associated with G 1. K u, A := { G u A u > 0 HA u = 0 Proof. It suffices to show that G u = G 1 u 1 for any u > 0. But this follows directly fro the uniqueness of weak liits, since 2.6 shows that Ktu, t G 1 u 1. A version of 2.6 unifor in u is needed for fdd convergence results. Proposition 2.3. Suppose K DG, and K is a tail kernel associated with G. Then, for any u > 0 and any non-negative function u t = ut such that u t u as t, we have 2.7 K tu t, t K u,, t. Proof. Suppose u t u > 0. Observe that Ktu t, t = Ktu t, tu t u 1 t, and put h t x = u t x, hx = ux. Writing P t = Ktu t, tu t, we have K tu t, t = P t h 1 t G h 1 = Gu 1 = K u, by 2, Theore 5.5, p. 34]. The easure G controls X upon leaving an extree state, and H describes the possibility of juping fro a non-extree state to an extree one. The traditional assuption 2.5 provides no inforation about H, and in fact 2.7 ay fail if u = 0 see Exaple 6.2. However, the choice of H cannot be ignored if 0 is an accessible point of the state space, especially for cases where G{0} = K y, {0} > 0. We propose pursuing iplications of the traditional assuption 2.5 alone, and will add conditions as needed to understand boundary behaviour of X. Alternative, ore general forulations of 2.5 include replacing Kt, t with Kt, at or Kt, at + bt with appropriate functions at > 0 and bt, in analogy with the usual doains of attraction conditions in extree value theory. Indeed, the second choice coincides with the original presentation by Perfekt 18], and relates to the conditional extree value odel 8, 13, 14]. For clarity, and to aintain ties with regular variation, we retain the standard choice at = t, bt = Representation. How do we characterize kernels belonging to DG? Fro 2.4, for chains transitioning according to a tail kernel, the next state is a rando ultiple of the previous one, provided the prior state is non-zero. We expect that chains transitioning according to K DG behave approxiately like this upon leaving a large state, and this is best expressed in ters of a function describing how a new state depends on the prior one. Given a kernel K, we can always find a saple space E, a easurable function ψ : 0, E 0, and an E-valued rando eleent V such that ψy, V Ky, for all y. Given a rando variable X 0, if we define the process X = X 0, X 1, X 2,... recursively as X n+1 = ψx n, V n+1, n 0, where {V n } is an iid sequence equal in distribution to V and independent of X 0, then X is a Markov chain with transition kernel K. Call the function ψ an update function corresponding to K. If in addition K DG, the doain of attraction condition 2.5 becoes t 1 ψt, V ξ,

5 ASYMPTOTICS OF MARKOV KERNELS AND THE TAIL CHAIN 5 where ξ G. Applying the probability integral transfor or the Skorohod representation theores 3, Theore 3.2, p. 6], 4, Theore 6.7, p. 70], we get the following result. Proposition 2.4. If K is a transition kernel, K DG if and only if there exists a easurable function ψ : 0, 0, 1] 0, and a rando variable ξ G on the unifor probability space 0, 1], B, λ such that 2.8 t 1 ψ t, u ξ u u 0, 1] as t, and ψ is an update function corresponding to K in the sense that λ ψ y, A ] = K y, A for easurable sets A. Think of the update function as ψ y, U where Uu = u is a unifor rando variable on 0, 1]. Proof. If there exist such ψ and ξ satisfying 2.8 then clearly K DG. Conversely, suppose ψ, V is an update function corresponding to K. According to Skorohod s representation theore cf. Billingsley 4] p. 70, with the necessary odifications to allow for an uncountable index set, there exists a rando variable ξ and a stochastic process {Yt ; t 0} defined on the unifor probability space 0, 1], B, λ, taking values in 0,, such that ξ G, Y 0 d = ψ0, V, Y t d = t 1 ψt, V for t > 0, and Yt u ξ u as t for every u 0, 1]. Now, define ψ : 0, 0, 1] 0, as ψ 0, u = Y0 u and ψ t, u = tyt u, t > 0, u 0, 1]. It is evident that λψ y, A] = Pψy, V A] for y 0,, so ψ is indeed an update function corresponding to K, and ψ satisfies 2.8 by construction. Update functions corresponding to K are not unique, and soe of the ay fail to converge pointwise as in 2.8. However 2.8 is convenient, and Proposition 2.4 shows that Segers 23] Condition 2.2 in ters of update functions is equivalent to our weak convergence forulation K DG. Pointwise convergence in 2.8 gives an intuitive representation of kernels in a doain of attraction. Corollary 2.1. K DG iff there exists a rando variable ξ G defined on the unifor probability space, and a easurable function φ : 0, 0, 1], satisfying t 1 φt, u 0 for all u 0, 1] such that 2.9 ψy, u := ξu y + φy, u is an update function corresponding to K. Proof. If such ξ and φ exist, then t 1 ψt, u = ξu + t 1 φt, u ξu for all u, so ψ satisfies 2.8. The converse follows fro 2.8. Many Markov chains such as ARCH, GARCH and autoregressive processes are specified by structured recursions that allow quick recognition of update functions corresponding to kernels in a doain of attraction. A coon exaple is the update function ψy, Z, W = Zy + W, which behaves like ψ y, Z = Zy when y is large copare ψ to the for 2.4 discussed for tail kernels. In general, if K has an update function ψ of the for 2.10 ψy, Z, W = Zy + φy, W for a rando variable Z 0 and a rando eleent W, where t 1 φt, w 0 whenever w C for which PW C] = 1, then K DG with G = PZ ]. We will refer to update functions satisfying 2.10 as being in canonical for.

6 6 S. I. RESNICK AND D. ZEBER 3. Finite-Diensional Convergence and the Extreal Coponent Given a Markov chain X K DG, we show that the finite-diensional distributions fdds of X, started fro an extree state, converge to those of the tail chain T defined in 2.4. We initially develop results that depend only on G but not H, and then clarify what behaviour of X is controlled by G and H respectively. We ake explicit links with prior work that did not consider the notion of boundary distribution. If G{0} = 0, the choice of H is inconsequential, since P T eventually hits {0} ] = 0 and T is indistinguishable fro the ultiplicative rando walk {Tn = T 0 ξ 1 ξ n, n 0} where T 0 > 0 and {ξ n } are iid G and independent of T 0. In this case, assue without loss of generality that H = ɛ 0. However, if G{0} > 0, any result not depending on H ust be restricted to fdds conditional on the tail chain not having yet hit {0}. For exaple, consider the trajectory of X 1,..., X, started fro X 0 = t, through the region t, 2 0, δ] t,, where t is a high level. The tail chain would odel this as a path through 0, 2 {0} 0,, which requires specifying H to control transitions away fro {0}. This raises the question of how to interpret the first hitting tie of {0} for T in ters of the original Markov chain X. Such hitting ties are iportant in the study of Markov chain point process odels of exceedance clusters based on the tail chain. Intuitively, a transition to {0} by T represents a transition fro an extree state to a non-extree state by X. We ake this notion precise in Section 3.2 by viewing such transitions as downcrossings of a certain level we ter the extreal boundary. We assue X is a Markov chain on 0, with transition kernel K DG, K is a tail kernel associated with G with unspecified boundary distribution H, and T is a Markov chain on 0, with kernel K. The finite-diensional distributions of X, conditional on X 0 = y, are given by and analogously for T. P y X1,..., X dx ] = K y, dx1 K x, dx2 K x 1, dx, 3.1. FDDs Conditional on the Intial State. Define the conditional distributions 3.1 π t X1 u, = Ptu t,..., X ] and π u, = Pu T1,..., T ], 1, t on 0, B0, ]. We consider when π t π on 0, ] pointwise in u. If G{0} = 0, this is a direct consequence of the doain of attraction condition 2.5, but if G{0} > 0, ore thought is required. We begin by restricting the convergence to the saller space E := 0, ] 1 0, ]. Relatively copact sets in E are contained in rectangles a, ] 0, ], where a 0, 1. Theore 3.1. Let u t = ut be a non-negative function such that u t u > 0 as t. a The restrictions to E, 3.2 µ t u, := π t u, E satisfy 3.3 µ t ut, v µ u, b If G{0} = 0, we have 3.4 π t ut, π u, and µ u, := π u, E, in M + E on 0, ] t. t. Proof. The Markov structure suggests an induction arguent facilitated by Lea 8.2 p. 21. Consider a first. If = 1, then 3.3 above reduces to 2.7. Assue 2, and let f C + K E.

7 ASYMPTOTICS OF MARKOV KERNELS AND THE TAIL CHAIN 7 Writing E = 0, ] E 1, we can find a > 0 and B KE 1 such that f is supported on a, ] B. Now, observe that µ t Defining h t v = ut, f = K tu t, tdx 1 0, ] E 1 = K tu t, tdx 1 0, ] E 1 E 1 K tx 1, tdx 2 K tx 1, tdx fx µ t 1 x1, dx 2,..., x fx. µ t 1 v, dx 1 fv, x 1 and hv = µ 1 v, dx 1 fv, x 1, E 1 the previous expression becoes ut, f = µ t Now, suppose v t v > 0 : we verify 0, ] 3.5 h t v t hv. K tu t, tdv h t v. By continuity, we have fv t, x t 1 fv, x 1 whenever x t 1 x 1, and the induction hypothesis provides µ t 1 v v t, µ 1 v,. Also, fx, has copact support B without loss of generality, µ 1 v, B = 0. Cobining these facts, 3.5 follows fro Lea 8.2 b. Next, since the h t and h have coon copact support a, ], and recalling fro Propostion 2.3 that Ktu t, t K u,, Lea 8.2 a yields ut, f K u, dv hv = µ u, f. µ t 0, ] Iplication b follows fro essentially the sae arguent. For 2, suppose f C0, ]. Replacing µ by π and E 1 by 0, ] 1 in the definitions of h t and h, we have π t ut, f = K tu t, tdv h t v. 0, ] This tie Lea 8.2 a shows that h t v t hv if v t v > 0, and since K u, 0, ] = 1, resorting to Lea 8.2 a once ore yields ut, f K u, dv hv = π u, f. π t 0, ] If G{0} > 0, then K u, 0, ] = 1 G{0} < 1, and for 3.4 to hold would require knowing the behaviour of h t v t when v t 0 as well. Behaviour near zero is controlled by an asyptotic condition related to the boundary distribution H. Previous work handled this using the regularity condition discussed in Section The Extreal Boundary. The noralization eployed in the doain of attraction condition 2.5 suggests that, starting fro a large state t, the extree states are approxiately scalar ultiples of t. For exaple, we would consider a transition fro t into t/3, 2t] to reain extree. Thus, we think of states which can be ade saller than tδ for any δ, if t is large enough, as non-extree. In this context, the set 0, t] would consist of non-extree states. Under 2.5, a tail chain path through 0, odels the original chain X travelling aong extree states, and all of the non-extree states are copacted into the state {0} in the state space of T. Therefore, if X is started fro an extree state, the portion of the tail chain depending solely on G is inforative up until the first tie X crosses down to a non-extree state. If G{0} = 0,

8 8 S. I. RESNICK AND D. ZEBER such a transition would becoe ore and ore unlikely as the initial state increases in which case G provides a coplete description of the behaviour of X in any finite nuber of steps following a visit to an extree state Theore 3.1 b. Drawing upon this interpretation, we develop a rigorous forulation of the distinction between extree and non-extree states, and we recast Theore 3.1 as convergence on the unrestricted space 0, ] of the conditional fdds, given that X has not yet reached a non-extree state. Definition. Suppose K DG. An extreal boundary for K is a non-negative function yt defined on 0,, satisfying li yt = 0 and 3.6 K t, t 0, yt] G{0} as t. Such a function is guaranteed to exist by Lea 8.5 p. 23. If G{0} = 0, then yt 0 is a trivial choice. For any function 0 yt 0, we have li sup Kt, t 0, yt] G{0}, so 3.6 is equivalent to 3.7 li inf K t, t 0, yt] G{0}. If yt is an extreal boundary, it follows that any function 0 ỹt 0 with ỹt yt for t t 0 is also an extreal boundary for K. Taking ỹt = s t ys shows that without loss of generality, we can assue yt to be non-increasing. The extreal boundary has a natural forulation in ters of the update function. As in 2.10, let ψy, Z, W = Zy + φy, W be an update function in canonical for, where y is extree. If Z > 0 then the next state is approxiately Zy, another extree state. Otherwise, if Z = 0, the next state is φy, W, and a transition fro an extree to a non-extree state has taken place. This suggests choosing an extreal boundary whose order is between t and φt, w. Proposition 3.1. Suppose ψy, Z, W is an update function in canonical for as in If ζt > 0 is a function on 0, such that 3.8 φt, w/ζt 0 as t whenever w B for which PW B] = 1, then li inf Kt, 0, ζt] G{0}. Provided li ζt/t = 0, an extreal boundary is given by yt := ζt/t. Thus if φt, w = oζt and ζt = ot then ζt/t is an extreal boundary. For exaple, if ψy, Z, W = Zy + W, so that φt, w = w, then choosing ζt to be any function ζt such that ζt = ot akes ζt/t an extreal boundary. Choosing ζt = t, we find that yt = 1/ t is an extreal boundary. Proof. Since we have P ψt ζt, Z = 0 ] = P φt, W ζt, Z = 0 ] P φt, W ζt, Z = 0 ] P Z = 0 ] ] φt, W P > 1 P Z = 0 ], ζt li inf K t, 0, ζt] = li inf P ψt ζt ] P Z = 0 ]. We will need an extreal boundary for which 3.6 still holds upon replacing the initial state t with tu t, where u t u > 0. Copare the following extension with Proposition 2.3. Proposition 3.2. If K DG, then there exists an extreal boundary y t such that 3.9 K tu t, t 0, y t] G{0} as t for any non-negative function u t = ut u > 0.

9 ASYMPTOTICS OF MARKOV KERNELS AND THE TAIL CHAIN 9 We will refer to y as a unifor extreal boundary. Proof. Let yt be an extreal boundary for K. As a first step, fix u 0 > 1, and suppose u 1 u < u 0. Define ỹt = u 0 ytu 1 0. Now, if u t u, then y {u} t := u t ytu t satisfies 3.9, since K tu t, t 0, y {u} t] = K tu t, tu t 0, ytu t ] G{0}. Here y {u} depends on the choice of function u t. However, since we eventually have u 1 0 < u t < u 0 for t large enough, it follows that ỹt > y {u} t for such t. Hence, ỹt satisfies 3.9 for any u t u with u 1 0 < u < u 0. Next, we reove the restriction in u 0 via a diagonalization arguent. For k = 2, 3,..., let y k t be extreal boundaries such that Ktu t, t 0, y k t] G{0} whenever u t u for u k 1, k, and put y 0 = y 1 = y. Next, define the sequence {s k, x k : k = 0, 1,... } inductively as follows. Setting s 0 = 0 and x 0 = y 0 1, choose s k s k 1 +1 such that y j t k 1 x k 1 for all j = 0,..., k whenever t s k, and put x k = ax{y j s k : j = 0,..., k}. Note that x k k 1 x k 1, so x k 0, and s k. Finally, set y t = x k 1 sk, s k+1 t. k=0 0 < Observe that 0 y t 0, and suppose u t u > 0. Then u k 1 0, k 0 for soe k 0, so Ktu t, t 0, y k0 t] G{0}, and for k k 0, our construction ensures that whenever s k t < s k+1, we have y k0 t y k0 s k x k = y t. Therefore, y t y k0 t for t s k0, so y satisfies 3.9. Henceforth, we assue any K DG is accopanied by a unifor extreal boundary denoted by yt, and we consider extree states on the order of t to be tyt, ]. If G{0} = 0, then all positive states are extree states. We now use the extreal boundary to reforulate the convergence of Theore 3.1 on the larger space 0, ]. Put E t = yt, ] 1 0, ], so that E t E = 0, ] 1 0, ]. Recall the notation µ t and µ fro 3.1, 3.2 in Theore 3.1 p. 6. Theore 3.2. Let u t = ut be a non-negative function such that u t u > 0 as t. Taking we have µ t µ t ut, v µ u, u, = π t u, E t, in M + 0, ] t. Proof. Note that we can just as well write µ t u, = µ t u, E t. Suppose 2 and let f C + K 0, ]. For δ > 0, define A δ = δ, ] 1 0, ], and choose δ such that µ u, A δ = 0. On the one hand, for large t we have µ t ut, f = fx 1 E tx µ t ut, dx 0, ] fx 1 Aδ x µ u, dx as t by Lea 8.3 p. 22. Letting δ 0 yields E E fx 1 Aδ x µ t ut, dx 3.10 li inf µt ut, f µ u, f

10 10 S. I. RESNICK AND D. ZEBER by onotone convergence. On the other hand, fixing δ, we can decopose the space according to the first downcrossing of δ: 3.11 µ t ut, f = ut, dx 1 + ut, dx, fx 1 Aδ x µ t 0, ] k=1 fx 1 A kx µ t 0, ] δ where A k δ = δ, ]k 1 0, δ] 0, ] k. On the subsets A k δ we appeal to the bound on f, say M, to obtain fx 1 A kx µ t 0, ] δ ut, dx M µ t ut, A k δ. Now, µ t ut, A k t 3.12 δ µ k ut, δ, ] k 1 yt, δ] = µ t k ut, δ, ] k 1 0, δ] µ t k ut, δ, ] k 1 0, yt]. Considering the second ter, we have ut, δ, ] k 1 0, yt] µ t k = = where 0, ] E k 1 K tu t, tdx 1 1δ, ] x 1 K tx k 2, tdx k 1 1δ, ] x k 1 K tx k 1, t 0, yt] 0, ] µ t k 1 ut, dx k 1 ht x k 1, Moreover, if x t k 1 x k 1 δ, ] k 1, then h t x k 1 = K tx k 1, t 0, yt] 1 δ, ] k 1x k 1. h t x t k 1 = K tx t k 1, t 0, yt] 1 δ, ] k 1x t k 1 G{0} 1 δ, ] k 1x k 1, using the fact that yt is a unifor extreal boundary. Since µ k 1 u, δ, ] k 1 = 0 without loss of generality by choice of δ, we conclude that ut, δ, ] k 1 0, yt] G{0} µ k 1 u, δ, ] k 1 = µ k u, δ, ] k 1 {0} µ t k as t. Now, let us return to Given any ɛ > 0, by choosing δ sall enough, we can ake ut, δ, ] k 1 yt, δ] µ k u, δ, ] k 1 0, δ] µ k u, δ, ] k 1 {0} µ t k µ k u, 0, ] k 1 0, δ] µ k u, δ, ] k 1 {0} < µ k u, 0, ] k 1 {0} + ɛ 2 µ k u, 0, ] k 1 {0} ɛ = ɛ, 2 i.e li sup µ t ut, A k δ < ɛ, for k = 1,..., 1. Therefore, 3.11 iplies that, given ɛ > 0, li sup µ t ut, f fx 1 Aδ x µ u, dx + M 0, ] < µ u, f + ɛ for sall enough δ. Cobining this with 3.10 yields the result. 1 k=1 li sup µ t ut, A k δ

11 ASYMPTOTICS OF MARKOV KERNELS AND THE TAIL CHAIN The Extreal Coponent. Having thus foralized the distinction between extree and non-extree states, we return to the question of phrasing a fdd liit result for X when H is unspecified. The extreal boundary allows us to interpret the first hitting tie of {0} by the tail chain as approxiating the tie of the first transition fro extree down to non-extree. In this terinology, Theore 3.2 provides a result, given that such a transition has yet to occur. Define the first hitting tie of a non-extree state τt = inf { n 0 : X n tyt }. For a Markov chain started fro tu t, where u t u > 0, we have tu t > yt for large t, so τt is the first downcrossing of the extreal boundary. For the tail chain T, put τ = inf{n 0 : T n = 0}. Given T 0 > 0, write τ = inf{n 1 : ξ n = 0}, where {ξ n } G are iid and independent of T 0, i.e. τ follows a Geoetric distribution with paraeter p = G{0}. Thus, Pτ = ] = p1 p 1 for 1 if p > 0, and Pτ = ] = 1 if p = 0. Theore 3.2 becoes 3.14 P tut t 1 X, τt ] v P u T, τ ], iplying that τ approxiates τt: 3.15 P tut τt ] P τ ], t, u t u > 0. So if G{0} > 0, X takes an average of approxiately G{0} 1 steps to return to a non-extree state. but if G{0} = 0, P tut τ 1 ] 0 for any 1 so starting fro a larger and larger initial state, it will take longer and longer for X to cross down to a non-extree state. Let T be the tail chain associated with G, ɛ 0. For {ξ n } G iid and independent of T 0, 3.16 T n = T 0 ξ 1 ξ n. We restate 3.14 in ters of a process derived fro X, called the extreal coponent of X, whose fdds converge weakly to those of T. The extreal coponent is the part of X whose asyptotic behavior is controlled by G alone. Definition. The extreal coponent of X relative to t is the process X t defined for t > 0 as X t n = X n 1 {n<τt}, n = 0, 1,.... Observe that X t is a Markov chain on 0, with transition kernel K t x, A { K x, A tyt, ] + ɛ 0 A K x, 0, tyt] x > tyt = ɛ 0 A x tyt. It follows that K t t, t G as t, and additionally that K t t, {0} G{0}. The relation between the coponent processes X t, T and the coplete ones is P tut t 1 X t τt > ] = P tut t 1 X τt > ] and P u T τ > ] = P u T τ > ]. Theore 3.3. Let u t = ut 0 satisfy u t u > 0 as t. Then on 0, ], π t ut, t ] X 1 := P tut t,..., Xt P u T t 1,..., T ] t.

12 12 S. I. RESNICK AND D. ZEBER Proof. Suppose 2 and f C0, ], and assue first that f 0. Then f C + K 0, ] as well, since the space is copact. Recall the notation of Theore 3.2. Conditioning on τt, we can write π t ut, f = = fx π t 0, ] fx π t 0, ] ut, dx + ut, dx + k=1 k=1 0, ] k 1 {0} fx π t 0, ] k 1 {0} k+1 ut, dx fx k, 0,..., 0 π t k ut, dx k by the Markov property. Since ut, 0, ] = P tut t 1 X t, τt > ] = P tut t 1 X yt, ] ] π t = µ t +1 ut, 0, ], the first ter becoes µ t ] +1 ut, f µ +1 u, f = fx π u, dx = fx P u T dx 0, ] 0, ] as t. Next, for any A 0, ] k easurable, write A 0 = {x k 1 : x k 1, 0 A} 0, ] k 1, and observe that π t k ut, A 0, ] k 1 {0} = P tut t 1 X t k 1 A 0 0, ] k 1, X t k = 0 ] = P tut t 1 X k 1 A 0 yt, ] k 1, t 1 X k yt ] = µ t k ut, A 0 0, ] µ t k+1 ut, A 0 0, ] 2. Applying this reasoning to the ters in the suation yields fx k 1, 0,..., 0 µ t 0, ]k+1 k ut, dx k fx k 1, 0,..., 0 µ t k+1 ut, dx k+1 0, ] k fx k 1, 0,..., 0 µ k u, dxk fx k 1, 0,..., 0 µ k+1 u, dxk+1 0, ] k 0, ] k+1 ] = fx k, 0,..., 0 π k u, dxk = fx P u T dx. 0, ] k 1 {0} k+1 0, ] k 1 {0} Cobining these liits shows that E tut f t 1 X t Eu ft, as t. Finally, if f is not non-negative, then write f = f + f. Since each of f + and f is non-negative, bounded, and continuous, we can apply the above arguent to each. 4. The Regularity Condition Previous work on the tail chain derives fdd convergence of X to T under a single assuption analogous to our doain of attraction condition 2.5. As we observed in Section 3.1, when G{0} = 0, fdd convergence of {t 1 X} follows directly, but when G{0} > 0, it was coon to assue an additional technical condition which ade 2.5 iply fdd convergence to T as well. This condition, which we refer to as the regularity condition, is an asyptotic convergence assuption prescribing the boundary distribution to be H = ɛ 0. We consider equivalences between different fors appearing in the literature, in ters of both kernels and update functions, and show that, under the regularity condition, the extreal behaviour of X is asyptotically the sae as that of its extreal coponent X t.

13 ASYMPTOTICS OF MARKOV KERNELS AND THE TAIL CHAIN 13 In cases where G{0} > 0, Perfekt 18, 19] requires that 4.1 li δ 0 li sup sup u 0,δ] K tu, t, ] = 0, while Segers 23] stipulates that the chosen update function corresponding to K ust be of at ost linear order in the initial state: 4.2 li sup sup t 1 ψy, v <, v B 0, PV B 0 ] = 1. 0 y t Sith 24] used a variant of 4.1. We dee a forulation in ters of distributional convergence to be instructive in our context. Definition. A Markov transition kernel K DG satisfies the regularity condition if 4.3 K tu t, t ɛ 0 on 0, ] as t for any non-negative function u t = ut 0. Note that in 2.7 p. 4, we had u t u > 0. We interpret 4.3 as designating the boundary distribution H to be ɛ 0. We now consider the relationships between 4.1, 4.2 and 4.3, and propose an intuitive equivalent for update functions in canonical for. Proposition 4.1. Suppose K DG, and let ψ, V be an update function corresponding to K such that 4.4 t 1 ψt, v ξv whenever v B for which PV B] = 1, and ξ V G. Then: a Condition 4.1 is necessary and sufficient for K to satisfy the regularity condition 4.3. b Condition 4.2 is sufficient for K to satisfy the regularity condition 4.3. c If ψ is in canonical for, i.e. ψy, Z, W = Zy + φy, W, then ψ satisfies 4.2 if and only if φ, w is bounded on any neighbourhood of 0 for each w C, a set for which PW C] = 1. Proof. a Assue 4.1, and suppose u t 0. We show Ktu t, tx, ] 0 for any x > 0. Write ωt, δ = sup K tu, t, ]. u 0,δ] Let ɛ > 0 be given, and choose δ sall enough that li sup ωt, δ < ɛ/2. Then for t large enough that u t < δx, we have K tu t, tx, ] sup K tu, tx, ] = ωtx, δ < li sup u 0,δx] ωt, δ + ɛ/2 for t large enough. Our choice of δ iplies that Ktu t, tx, ] < ɛ. Conversely, assue that K satisfies 4.3 but that 4.1 fails. Choose ɛ > 0 and a sequence δ n 0 such that li sup ωt, δ n ɛ for n = 1, 2,.... Then for each n we can find a sequence t n k as k such that ωtn k, δ n ɛ for each k. Diagonalize to find k 1 < k 2 < such that s n = t n k n and ωs n, δ n ɛ for all n. Finally, for n = 1, 2,... choose u n 0, δ n ] such that K s n u n, s n, ] > ωs n, δ n ɛ/2, and put ut = n u n 1 sn,s n+1 t. Clearly ut 0, but Ks n us n, s n, ] ɛ/2 for all n, contradicting 4.3.

14 14 S. I. RESNICK AND D. ZEBER b Write Mv = li sup t sup 0 y t t 1 ψy, v. Since for δ > 0, we have sup t 1 ψtδ 1 y, v ψy, v = sup 0 y t 0 y δ tδ 1 δ 1 li sup sup 0 y δ Now, suppose u t 0. Given any δ > 0 we have t 1 ψty, v = δmv. t 1 ψtu t, v sup t 1 ψty, v 0 y δ provided t is large enough, so li sup t t 1 ψtu t, v δmv. Consequently, li sup t t 1 ψtu t, v = 0 for every v such that Mv <. Under 4.2, this eans that P t 1 ψtu t, V 0 ] = 1, iplying 4.3. c Suppose first that χ w a = sup 0 y a φy, w < for all a > 0, whenever w C. Fixing w C and z 0, note that and observe for any a > 0 that sup 0 y t t 1 φy, w sup 0 y a t 1 φy, w sup t 1 ψy, z, w z + sup t 1 φy, w, 0 y t 0 y t sup a y t Choosing a large enough that sup a y y 1 φy, w 1, say, it follows that li sup sup 0 y t y 1 φy, w t 1 χ w a sup y 1 φy, w a y. t 1 ψy, z, w z + 1, so v = z, w B 0. Therefore PZ, W B 0 ] PZ 0, W C] = 1. Conversely, suppose there is a set D with PW D] > 0 such that w D iplies χ w a = for soe 0 < a <. Since sup 0 y t t 1 ψy, z, w t 1 χ w t, we have 0, D B0 c, contradicting 4.2. The exclusion of necessity fro part b results fro the fact that a kernel K does not uniquely specify an update function ψ. Even when K satisfies the regularity condition 4.3, it ay be possible to choose a nasty update function ψ which satisfies 4.4, but not 4.2. However, in such cases there ay exist a different update function ψ corresponding to K which does satisfy 4.2. Here is an exaple of such a situation. We exhibit an update function ψ for which i 4.4 holds; ii 4.2 fails because condition c in Proposition 4.1 fails; but yet iii the corresponding kernel satisfies the regularity condition 4.3. Furtherore, we present a different choice of update function corresponding to the sae kernel which satisfies 4.2. Define ψy, V = Z, W = Zy + φy, W, where φy, w = k 1 {yw=1/k} k=1 and W U0, 1. i Since φt, w = 0 for t > 1/w, it is clear that ψ satisfies 4.4 with ξ = Z. ii Observe that for any w 0, 1, φ, w is unbounded on the interval 0, 1]. Therefore, by part c of Proposition 4.1, 4.2 cannot hold for ψ. iii However, the corresponding kernel does satisfy the regularity condition 4.3. Suppose u t 0 and a > 0 is arbitrarily large. Write P t 1 ψtu t, Z, W > x ] = P Zu t + t 1 φtu t, W > x ] P t 1 φtu t, W > x ] + PZ > a],

15 ASYMPTOTICS OF MARKOV KERNELS AND THE TAIL CHAIN 15 choosing 0 < x < x au t. Since for any t, {w : φtu t, w > tx } {tu t k 1 : k = 1, 2,... }, a set of easure 0 with respect to PW ], 4.3 follows by letting a. On the other hand, the update function ψ y, Z = Zy does satisfy 4.2, and for any y, P ψ y, Z ψy, Z, W ] = P W {yk 1 : k = 1, 2,... } ] = 0, so ψ does indeed correspond to K. The regularity condition 4.3 restricts attention to Markov chains for which the probability of returning to an extree state in the next steps after falling below the extreal boundary is asyptotically negligible. For such chains, as well as those for which yt 0 is an extreal boundary for K, X has the sae asyptotic behaviour as its extreal coponent, as described next. Theore 4.1. Suppose X K with K DG, and let ρ be a etric on R. If yt 0 is an extreal boundary for K, or if K satisfies the regularity condition 4.3, then for any ɛ > 0 we have t X 4.5 P tut ρ t Consequently, X1 4.6 P tut t,..., X t, X ] > ɛ 0 t, u t u > 0. t ] P u T 1,..., T ] t, u t u > 0. First let us extend the regularity condition to higher-order transition kernels. Lea 4.1. If K satisfies 4.3, then so do the -step transition kernels K. Proof. This is established by induction. Let u t 0 and f C0, ]. For 2, we have K tu t, f = K 1 tu t, tdv K tv, tdx fx. 0, ] 0, ] Assue that K 1 tu t, t ɛ 0 ; 4.3 iplies that Ktv t, tdx fx f0 whenever v t 0. Therefore, by Lea 8.2 a p. 21, we conclude that K tu t, f f0 = ɛ 0 f. Proof of Theore 4.1. Suppose ɛ > 0 and u t u > 0. Write P tut ρt 1 X t, t 1 X > ɛ ] = P tut ρt 1 X t, t 1 X > ɛ, τt = k ]. Since X j = X t j k=1 while j < τt, for the k-th suand to converge to 0, it is sufficient that P tut X t j /t X j /t > δ, τt = k ] = P tut Xj /t > δ, τt = k ] 0 for j = k,..., and any δ > 0. If j = k, we have P tut Xj /t > δ, τt = k ] P tut Xk /t > δ, X k /t yt ] = 0 for large t. For j > k, recalling the notation of Theore 3.2, P tut Xj /t > δ, τt = k ] = 1 0,yt] x k P tut Xj /t > δ ] ] X k /t = x k Ptut Xk /t dx k E k t = P txk Xj k > tδ ] 1 0,yt] x k µ t k ut, dx k 0, ] k

16 16 S. I. RESNICK AND D. ZEBER using the Markov property. We clai that this intergral 0 as t. If yt 0, this follows directly. Otherwise, recall that µ t k u t, v µ k u,, and consider h t x k = P txk X j k > tδ] 1 0,yt] x k. Suppose x t x 0, ] k. If x k > 0, then h t x t = 0 for large t because yt 0. Otherwise, if x k = 0, we have h t x t 0 since Lea 4.1 iplies that P t tx X j k > tδ] 0 k as t. Lea 8.2 b establishes 4.5; 4.6 follows by Slutsky s theore. Therefore, X converges to T in fdds under a G{0} = 0, b G{0} > 0 cobined with 4.3, or c G{0} > 0 cobined with the extreal boundary yt 0. In either case, we will be able to replace the extreal coponent X t with the coplete chain X in the results of Sections 5.1 and 5.2. However, that yt 0 is an extreal boundary, and consequently that 4.6 holds, does not iply the regularity condition to hold, regardless of G{0}; in particular, a kernel for which G{0} = 0 need not satisfy 4.3. This is illustrated in Exaple Convergence of the Unconditional FDDs 5.1. Effect of a Regularly Varying Initial Distribution. So far our convergence results required that the initial state becoe large, and the only distributional assuption was that the transition kernel K deterining X be attracted to soe distribution G. To obtain a result for the unconditional distribution of X 0,..., X, we require an additional assuption about how likely the initial observation X 0 is to be large. Using Lea 8.4, the results of the previous sections extend to ultivariate regular variation on the cone E = 0, ] 0, ] when the distribution of X 0 has a regularly varying tail. This cone is saller than the cone 0, ] +1 \{0} traditionally eployed in extree value theory, because the kernel doain of attraction condition 2.5 is uninforative when the initial state is not extree. This is analogous to the setting of the Conditional Extree Value Model considered in 8, 13]. Proposition 5.1. Assue X K with K DG, and X 0 H, where H is a distribution on 0, with a regularly varying tail. This eans that as t, for soe scaling function bt, th bt v ν α in M + 0, ], where ν α x, ] = x α and α > 0. Define the easure ν on E = 0, ] 0, ] by 5.1 ν dx 0, dx = να dx 0 P x0 T 1,..., T dx ]. Then, for = 1, 2,..., the following convergences take place as t : a In M + 0, ] 0, ], t P bt 1 X 0, X 1,..., X 0, ] 0, ] ] v ν 0, ] 0, ]. b In M + E, t P bt 1 X bt 0, X bt 1,..., X bt ] v ν. c If either G{0} = 0, yt 0 is an extreal boundary, or K satisfies the regularity condition 4.3, then in M + E, t P bt 1 X 0, X 1,..., X ] v ν. d In M + 0, ], t P X 0 /bt dx 0, τbt ] v 1 G{0} 1 να dx 0. Reark. These convergence stateents ay be reforulated equivalently as, say, P bt 1 X 0, X 1,..., X X0 > bt ] P T 0, T 1,..., T ], where T0 Paretoα. This is the for considered by Segers 23].

17 ASYMPTOTICS OF MARKOV KERNELS AND THE TAIL CHAIN 17 Proof. Apply Lea 8.4 p. 22 to the results of Theores 3.1, 3.3 and 4.1, and In the case = 1, E 1 is a rotated version of E used in the conditional extree value odel in 8, 9] and the liit can be expressed as ν x 0, ] 0, x 1 ] = x 0 ν α dup ξ x 1 /u] = x α 0 P ξ x 1 /x 0 ] x α 1 Eξ α 1 {ξ x1 /x 0 } for x 0, x 1 0, ] 0, ], where ξ G with Eξ α. Since ν x 0, ] {0} = x α 0 Pξ = 0] and ν 0, ] x 1, ] = x α 1 Eξ α, sets on the x 0 -axis incur ass proportional to G{0}, and sets bounded away fro this axis are weighted accordng to Eξ α. A consequence of the second observation is that li inf t P X 1 /bt > x ] Eξ α x α. Thus, knowledge concerning the tail behaviour of X 1 iposes a restriction on the distributions G v to which K can be attracted via the α-th oent. For exaple, if t PX 1 /bt ] ν α, then we ust have Eξ α 1; this property will be exained further in the next section and appears in various fors in Segers 23] and Basrak and Segers 1], in the stationary setting Joint Tail Convergence. What additional assuptions are necessary for convergences b and c of the previous result to take place on the larger cone E = 0, ] +1 \{0}? This was considered by Segers 1, 23] for stationary Markov chains. In b, the dependence on the extreal threshold and hence on t eans we are in the context of a triangular array and not, strictly speaking, in the setting of joint regular variation. However, the result is still useful, for exaple, to derive a point process convergence via the Poisson transfor 21, p. 183]. As a first step, we characterize convergence on the larger cone by decoposing it into saller, ore failiar cones. This is siilar to Theore 6.1 in 23] and one of the iplications of Theore 2.1 in 1]. As a convention in what follows, set 0, ] 0 A = A. Also, recall the notation E = 0, ] 0, ]. Proposition 5.2. Suppose Y t = Y t,0, Y t,1,..., Y t, is a rando vector on 0, ] +1 for each t > 0. Then there exists a non-null Radon easure µ on E = 0, ] +1 \{0} such that 5.2 t P Y t,0, Y t,1,..., Y t, ] v µ in M + E t if and only if for j = 0,..., there exist Radon easures µ j on E j = 0, ] 0, ] j, not all null, such that 5.3 t P Y t,j,..., Y t, ] v µ j in M + E j. The relation between the liit easures is the following: for j = 0,...,, and µ 0, x] c = µ j = µ 0, ] j on E j µ j xj, ] 0, x j+1 ] 0, x ] for x E. j=0 Furtherore, given j {0,..., 1}, if A 0, ] j \{0} j µ j 0, ] A <. is relatively copact, then

18 18 S. I. RESNICK AND D. ZEBER Proof. Assue first that 5.2 holds. Fixing j {0,..., }, define µ j := µ 0, ] j i.e. µ = µ. Let A E j be relatively copact with µ j A = 0. Then A = 0, ] j A is relatively copact in E, and E A = 0, ] j E j A, so µ E A = µ j A = 0. Therefore, t P Y t,j,..., Y t, A ] = t P Y t,0,..., Y t, A ] µ A = µ j A, establishing 5.3. Conversely, suppose we have 5.3 for j = 0,...,. For x 0, ] +1, define hx = µ j xj, ] 0, x j+1 ] 0, x ]. j=0 Decopose 0, x] c as a disjoint union 5.4 0, x] c = 0, ] j x j, ] 0, x j+1 ] 0, x ], j=0 and observe that at points of continuity of the liit, 5.5 t P Y t 0, x] c ] = t P Y t,j,..., Y t, x j, ] 0, x j+1 ] 0, x ] ] hx. j=0 Hence, 5.2 holds with the liit easure µ defined by µ 0, x] c = hx. Indeed, given f C + K E we can find δ > 0 such that x δ = δ,..., δ is a continuity point of h and f is supported on 0, x δ ] c. Therefore, t EfY t sup fx sup t P Y t 0, x δ ] c] <, x E t>0 iplying that the set { t PY t ] ; t > 0 } is relatively copact in M + E. Furtherore, if t k PY tk ] µ and s k PY sk ] µ as k, then µ = µ = µ on sets 0, x] c which are continuity sets of µ by 5.5. This extends to easurable rectangles in E bounded away fro 0 whose vertices are continuity points of h, leading us to the conclusion that µ = µ = µ on E. Moreover, since we can decopose 0, x] c for any x E as in 5.4, it is clear that µ is non-null iff not all of the µ j are null. Finally, for 1 j 1, if A 0, ] j \{0} j is relatively copact, then it is contained in 0,..., 0, x j+1,..., x ] c for soe x j+1,..., x 0, ] j. Applying 5.4 once again, we find that µ j 0, ] A = µ 0, ] j 0, ] A µ 0, ] j+1 0, ] k j 1 x k, ] 0, x k+1 ] 0, x ] = k=j+1 k=j+1 µ k xk, ] 0, x k+1 ] 0, x ] <. Consequently, the extension of the convergences in Proposition 5.1 to the larger cone E follows fro regular variation of the arginal tails. Theore 5.1. Suppose X K DG, and let bt be a scaling function and α > 0. Then 5.6 t P bt 1 X bt 0, X bt 1,..., X bt ] v µ in M + E t,

19 where if and only if ASYMPTOTICS OF MARKOV KERNELS AND THE TAIL CHAIN 19 µ E dx 0, dx = ν α dx 0 P x0 T 1,..., T dx ] = ν dx 0, dx, 5.7 t P X bt j /bt ] v c j ν α in M + 0, ], with c 0 = 1 and Eξ α j c j < for j = 1,...,. Proof. Assue first that 5.6 holds. It follows that t P X 0 > btx] ν x, ] 0, ] = x α for x > 0. Hence, bt RV 1/α, so by 5.6 again, we have for j 1 and t P X bt j > btx ] µ 0, ] j x, ] 0, ] j = c j x α, c j µ 0, ] 0, ] j 1 1, ] 0, ] j = = Eξ 1 ξ j α = Eξ α j. 0, ] ν α du P ξ 1 ξ j > u 1 ] Conversely, suppose that 5.7 holds for j = 0,...,. Lea 8.4 iplies that in M + E j, t P bt 1 X bt j,..., X bt dx 0, dx ] v c j ν α dx 0 P x0 T 1,..., T j dx ] =: µ j dx0, dx by the Markov property, and Proposition 5.2 yields 5.6, with µ E = µ = ν. At the end of Section 4, cases were outlined in which we could replace X bt j by X j. Theore 5.1 is ost striking for these since it shows that for a Markov chain whose kernel is in a doain of attraction, to obtain joint regular variation of the fdds it is enough to know that the arginal tails are regularly varying. In particular, if X has a regularly varying stationary distribution then the fdds are jointly regularly varying. This result was presented by Segers 23], and Basrak and Segers 1] showed that for a general stationary process, joint regular variation of fdds is equivalent to the existence of a tail process which reduces to the tail chain in the case of Markov chains. However, what Proposition 5.1 ephasizes is that it is the arginal tail behaviour alone, rather than stationarity, which provides the link with joint regular variation. Theore 5.1 also extends the observation ade in Section 5.1 that knowledge of the arginal tail behaviour for a Markov chain whose kernel is in a doain of attraction constrains the class of possible liit distributions G via its oents. If a particular choice of regularly varying initial v distribution leads to t PX j > bt ] a j ν α, then we have Eξ α a 1/j j. In particular, if X adits a stationary distribution whose tail is RV α, then Eξ α 1. Our first exaple illustrates the ain results. 6. Exaples Exaple 6.1. Let V = Z, W be any rando vector on 0, R. Consider the update function ψy, V = Zy + W + and its canonical for ψy, V = Zy + φy, W = Zy + W 1 {W > Zy} Zy 1 {W Zy}. For y > 0, the transition kernel has the for Ky, x, = P Zy + W > x]. Since t 1 ψt, V = Z +t 1 W + Z a.s., we have K DG with G = PZ ]. Furtherore, using Proposition 3.1, the function γt t is of larger order than φt, w, so yt = 1/ t is an extreal boundary. Since

20 20 S. I. RESNICK AND D. ZEBER φ, w is bounded on neighbourhoods of 0, Proposition 4.1 c iplies K satisfies the regularity condition 4.3. Consequently, fro Theore 4.1, we obtain fdd convergence of t 1 X to T as in 4.6. If K does not satisfy the regularity condition 4.3, Theore 4.1 ay fail to hold and starting fro tu, t 1 X ay fail to converge to T started fro u. Exaple 6.2. Let V = Z, W, W be any non-degenerate rando vector on 0, 3, and consider the Markov chain deterined by the update function ψy, V = Zy + W y 1 1 {y>0} + W 1 {y=0}. For y > 0, the transition kernel is Ky, x, = PZy + W y 1 > x] and since t 1 ψt, V = Z + W t 2 Z a.s., we have K DG with G = PZ ]. Furtherore, using Proposition 3.1, the function γt 1 is of larger order than φt, w, so yt = 1/t is an extreal boundary. However, note that φy, W, W = W y 1 1 {y>0} +W 1 {y=0} is unbounded near 0, iplying that Segers boundedness condition 4.2 does not hold. In fact, our for of the regularity condition 4.3 fails for K. Indeed, K tu t, tx, = PZtu t + W/tu t > tx] = PZu t + W/t 2 u t > x]. Choosing u t = t 2 yields Ktu t, tx, PW > x]. For appropriate x, this shows 4.3 fails. Not only does 4.3 fail but so does Theore 4.1, since the asyptotic behaviour of X is not the sae as that of X t. We show directly that the conditional fdds of t 1 X fail to converge to those of T. The idea is that if X k < yt = t 1, there is a positive probability that X k+1 > t. We illustrate this for = 2. Take f C0, ] 2 and u > 0. Observe if X 0 = tu > 0, fro the definition of ψ, X 1 = Z 1 tu + W 1 /tu and X 2 = Z 2 X 1 + W 2 /X 1 1 {X1 >0} + W 1 {X1 =0}. Furtherore, on {Z 1 > 0}, we have X 1 > 0 and X 2 = Z 2 X 1 +W 2 /X 1. On {Z 1 = 0, W 1 > 0}, X 1 > 0 and X 2 = Z 2 X 1 +W 2 /X 1. On {Z 1 = 0, W 1 = 0}, we have X 1 = 0 and X 2 = W. Therefore E tu fx 1 /t, X 2 /t = E tu fx 1 /t, X 2 /t 1 {Z1 >0} + E tu fx 1 /t, X 2 /t 1 {Z1 =0,W 1 >0} + E tu fx 1 /t, X 2 /t 1 {Z1 =0,W 1 =0} = A + B + C. For A, as t, we have A = Ef Z 1 u + W 1 /t 2 u, Z 2 Z 1 u + W 1 /t 2 u] + W 2 /Z 1 t 2 u + W 1 u 1 ] 1 {Z1 >0} EfZ 1 u, Z 1 Z 2 u 1 {Z1 >0}, while for B we obtain for t, B = EfW 1 /t 2 u, Z 2 W 1 /t 2 u + W 2 u/w 1 1 {Z1 =0,W 1 >0} Ef0, uw 2 /W 1 1 {Z1 =0,W 1 >0}. Finally for C, C = Ef0, W 2/t 1 {Z1 =0,W 1 =0} = PZ 1 = 0, W 1 = 0] Ef0, W 2/t PZ 1 = 0, W 1 = 0] f0, 0. Observe that li A + B + C] E u ft 1, T 2 = EfuZ 1, uz 1 Z 2. In the final exaple, the conditional distributions of t 1 X converge to those of the tail chain T, even though the regularity condition does not hold. This includes cases for which G{0} = 0 and G{0} > 0 with extreal boundary yt 0. Exaple 6.3. Let {ξ j, η j, j 1} be iid copies of the non-degenerate rando vector ξ, η on 0, 2. Taking V = ξ, η, consider a Markov chain which transitions according to the update function ψy, V = ξ y + y 1 1 {y>0} + η 1 {y=0} = ξ y + ξ y 1 1 {y>0} + η 1 {y=0},