Nonparametric Estimation in Null Recurrent Time Series

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NONPARAMETRIC ESTIMATION IN NULL RECURRENT TIME SERIES

B

^Y

H

^ANS

A

^RNFINN

K

^ARLSEN ^AND

D

^AG

T

^J ^STHEIM

University of Bergen

Department of Mathematics, Johannes Bruns gt.12, 5007 Bergen, Norway 17th May 1998

Abstract

We develop a nonparametric estimation theory in a nonstationary environment, more precisely in the framework of null recurrent Markov chains. An essential tool is the split chain, which makes it possible to decompose the times series under consideration in independent and identical parts. A tail condition on the distribution of the recurrence time is introduced. This condition makes it possible to prove weak convergence results for series of functions of the process depending on a smoothing parameter. These limit results are subsequently used to obtain consistency and asymptotic normality for local density estimators and for estimators of the conditional mean and the conditional variance. In contra- distinction to the parametric case, the convergence rate is slower than in the stationary case, and it is directly linked to the tail behaviour of the recurrence time.

Key words andphrases. Nonstationary time series models, null recurrent Markov chain, nonparametric kernel estimators, split chain.

AMS1991subjectclassication. Primary 62M10, 62G07 secondary 60J05.

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1 Introduction

Work on nonparametric estimation has so far with very few exceptions been carried out in a stationary strongly mixing framework (see e.g. Robinson, 1983, Masry and Tjstheim, 1995, and references therein). Recently asymptotics for processes with long range dependence have been covered (Robinson, 1997), but still no systematic theory exists for a nonstationary situation.

The main purpose of this paper is to try to ll this gap by establishing a nonparametric estimation theory that can be used in a nonstationary environment. Clearly the collection of all nonstationary processes is much too wide, but in our opinion an appro- priate framework for working with such problems is the class of null recurrent Markov chains, or possibly regime models including null recurrent states. It is true that this requires the model to be stated as a Markov chain, but this is a mild restriction. The random walk model and many of the related unit-root processes belong to this class (Myklebust et al, 1998a), and, more importantly, nonlinear processes are not excluded.

With the single exception of the work by Yakowitz (1993) on consistency of nearest neighbour estimates, as far as we know, the estimation theory of null recurrent processes has been conned to the parametric case. Asymptotics of parametric (usually non-time series) models have been treated by Hpfner (1990, 1994), Hpfner et al (1990), Kasahara (1982, 1984, 1985), Touati (1990), and we will exploit some of their techniques. For two early contributions in this eld we refer to Darling & Kac (1957) and Kallianpur & Robbins (1954). However, there are important dierences between the parametric and nonparametric situations. A parametric estimate is strongly inu- enced by the large values of the process, and for unit-root processes super-eciency is obtained with a faster rate of convergence than in the stationary case. In contradistinc- tion, a nonparametric estimator depends heavily on observations which are conned to a neighbourhood of a given point, and the rate of convergence turns out, not unexpec- tedly, to be slower than in the stationary case. This means that series with large or very large sample sizes are required.

Long series are becoming increasingly available, e.g. in nance and econometrics.

There is therefore also a practical motivation behind our work. The particulars of this motivation are much the same as for the stationary case: it is desirable to have greater exibilityin the initial stage of modelling than that oered by a xed parametric or semiparametric model, for example using nonparametric estimates as a guide in choosing a parametric (linear or nonlinear) model. Since the present paper is directed towards establishing a theory, specic practical aspects are not discussed, and we refer to Myklebust et al (1998a) for some examples and details on practical implementations.

We would like to mention very briey potential implications for econometric time series modelling, though, since such series are often thought to be nonstationary. The kind of nonstationarity that has been built into the parametric econometric modelling has overwhelming been of linear unit-root type leading to ARIMA models and, in the multivariate case, to linear cointegration models. For such models a very considerable body of literature exists (cf. the review papers by Stock, 1994, Watson, 1994 and the book by Johansen, 1995). Asymptotic distributions are typically non-normal and the parameter estimates are super-ecient (Dickey & Fuller, 1979, Johansen, 1995).

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4 1 INTRODUCTION The need for models combining features of nonlinearity and nonstationarity has been emphasized (see e.g. Granger and Hallman, 1991, Granger, 1995, Aparicio & Escribano, 1997), but once more no systematic estimation theory exists. Again, we believe that the class of null recurrent processes constitutes an adequate framework for posing such problems. The technique used in this paper is general, and although we focus on nonparametric estimation, it is in principle possible to develop an analogous theory covering nonlinear and nonstationary parametric time series models. Finally, it should be mentioned that there are challenging and interesting connections to attempts having been made to construct a nonlinear cointegration theory. We look at some of these in Myklebust et al (1998a).

There are a number of open problems and possibilities for further research. These are related to exploratory problems such as those examined by Tjstheim and Auestad (1994), Masry and Tjstheim (1997) and Hjellvik et al (1998), but there are also many hard problems connected with the basic estimation theory itself. A few of the latter ones are looked at in Myklebust et al (1998b).

Since our paper draws quite heavily on Markov theory for recurrent chains, we start in Section 2 by stating some main facts stemming from that theory. Much of the material is based on the book by Nummelin (1984), but since, to our knowledge, it has not been utilized before in the context of nonparametric estimation, it has been included to make the paper more self-contained. In fact, we consider the merger of the recurrence theory of Markov chains in particular use of the split chain and the asymptotictheory of sums depending on a smoothing parameter to be a main contribution of the paper.

This synthesis is achieved in Section 3. Applications to nonparametric estimation of an invariant density and conditional mean/variance functions are given in Section 4, where we derive consistency and asymptotic normality of these estimates in a null recurrent situation. Some details of the technical derivations are relegated to two appendices.

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5

2 Markov theory

2.1 Notation

We adopt the notation used by Nummelin (1984). We denote by ^fXt t 0^g a - irreducible Markov chain on a general state space (E^E) with transition probabilityP.

The sigma algebra of measurable sets,Ê, is countably generated and we assume that is maximal in the sense that if ⁰ is another irreducible measure then ⁰ is absolutely continuous with respect to. We denote the class of non-negative measurable functions with-positive support byÊ⁺. For a setA²Ê we writeA²Ê⁺ if the indicator function 1A ²Ê⁺. The chain is Harris recurrent if for all A²Ê⁺

P(SA <¹^jX⁰ =x)1 where SA = min^fn 1:Xn²A^g: (2:1) In the following^fXt t 0^gwill always be assumed to be-irreducible Harris recurrent.

The chain is positive recurrent if there exists an initial probability measure such that

fXt t 0^g is strictly stationary, and the process is null recurrent otherwise.

If is a non-negative measurable function and is a measure, then the kernel is dened by

(xA) = (x)(A) (xA)²(E^E):

If K is a general kernel, the function K, the measure K and the number are dened by

K(x) =^Z K(xdy)(y) K(A) = ^Z (dx)K(xA) =^Z (dx)(x) : Sometimes we write() instead of . The convolution of two kernels K¹ andK² gives another kernel dened by

K¹K²(xA) = ^Z K¹(xdy)K²(yA) :

Due to associative laws the number K¹K² is uniquely dened. If A ² ^E and 1A

is the corresponding indicator variable, then K1A(x) = K(xA). The kernel I is dened by I (xA) = (x)1A(x) (and I (xdy) = (x)x(dy) where x is the Dirac Delta measure at the point x). We abbreviate the identity function 1E by 1. We let

Gdr = ^ff: (E^r^E^r) ^7! (R^d^B(R^d)^g where ^B(R^d) is the class of Borel sets on R^d. If r = 1 or d = 1, we drop the subscript or superscript.

We dene a ² ^E⁺ to be small if there exists a measure , a positive constant b and an integerm 1 so that

P^m b : (2:2)

If satises (2.2) for some , b and m, then is a small measure.

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6 2 MARKOV THEORY

2.2 Basic conditions and the split chain

A fundamental fact for -irreducible Markov chains is the existence of a minorization inequality (Nummelin 1984, Th. 2.1 and Prp. 2.6, pp. 16-19): there exists a small function s, a probability measure and an integer m⁰ 1 so that

P^m⁰ s : (2:3)

It creates some technical diculties to have m⁰ > 1, and it is not a severe restriction to assume m⁰ = 1. Therefore, unless otherwise is stated, in the sequel we assume that the minorization inequality

P s (2:4)

holds, where s and are small and (E) = 1. In particular, this implies that 0 s(x)1 x²E. If (2.4) holds, then the pair (s) is called an atom (for P).

We illustrate what the minorization inequality means in the case of a nonlinear autoregressive process:

Example 2.1

Assume that

Xt= X⁰ ^when t = 0 f(Xt^;1) +Zt when t 1

where ^fZt t 0^g are iid random variables with zero mean and with density ^with respect to the Lebesgue measure on E = R. Assume that the function f is bounded on compact sets and infx²C(x) is strictly positive for all compact sets C. The transition probability is given by

P(xdy) = p⁽¹⁾(y^jx)dy^def= (y^;f(x))dy and the n step transition function is Pⁿ(xdy) = p⁽ⁿ⁾(y^jx)dy ^where

p⁽ⁿ⁾(y^jx) =^Z p⁽ⁿ^;1)(y^ju)(u^;f(x))du n 2 : (2:5) Let C be a compact set with positive Lebesgue measure. Dene ⁰(y) = infx²C(y ^; f(x))^, a =^R ⁰(y)dy^, = a^;1⁰^, s = a1C. Then

P(xdy) 1C(x)⁰(y)dy

= s(x)(dy)

where (dy) = (y)dy and (E) = 1. Thus (2.4) is satised .

In the nonparametric estimation theory an important role will be played by the split chain, which can be constructed once the minorization condition is fullled. It permits splitting the chain into separate and identical parts which are building blocks in the subsequent analysis.

We introduce an auxiliary chain ^fYt^g, where Yt can only take the values 0 and 1.

The split chain,^fXtYtt 0^gis dened on an extension of the basic probability space

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2.2 Basic conditions and the split chain 7 so that = E ^f1^g is a proper atom. The simplest description of this construction is given by an algorithm. Let E = Eê ^f01^g and Êê the corresponding extension of Ê. Dene

y(x) = s(x)y + (1^;s(x))(1^;y) = s(x) y = 1

1^;s(x) y = 0 (2:6) with s as in (2.4). For each xed y ² ^f01^g, y is a function dened on E. For an arbitrary measure and an arbitrary function f, dened on (EÊ) let andê f denoteê the extension to (Eê Êê) given by

f(xy) = f(x)e y(x) (dxe ^fy^g) = (dx)y(x) : Let

Q(xA) =1^;s(x)^;1P(xA)^;s(x)(A)1(s(x) < 1) + 1A(x)1(s(x) = 1) and y(xA) = (A)y + Q(xA)(1^;y) = (A) y = 1

Q(xA) y = 0. (2:7)

By (2.4) the kernel y is a probability kernel on (E^E) for each xedy.

Let denote an arbitrary initial distribution on E , let ^F^;1^Y be the trivial sigma algebra and dene ^f(XtYt)t 0^g by

P(X⁰ ²A) = (A)

P(Yt=y ^j^F_t^X ^_^F_Yt^;1) = y(Xt) t 0

P(Xt²A^j^Ft^X^;1^_^FYt^;1) = Yt^;1(Xt^;1A) t 1 (2:8) where ^Ft^X and ^FYt are the -algebras generated by ^fXj j t^g and ^fYj j t^g. We observe that the distribution of ^f(XnYn) n 0^g is determined by , P and (s) . We use IP as generic symbol for the distribution of the Markov chain with initial distribution, and the corresponding expectation is denoted by IE. If the actual expressions involved are independent of, then we may drop the subscript.

Lemma 2.1

The split chain dened by (2.8) is a Markov chain with state space Eê^, initial distribution ê and transition probability function Pê given by

Pe(x⁰y⁰)(dx^fy^g)= y⁰(x⁰dx)y(x) : (2:9) The set = E ^f1^g is a proper atom for this chain i.e., P((xy)^e )is independent of (xy) ^when (xy) ² ^{. The} X- marginal process of the compound chain has the same properties as the original chain, moreover

IP(Xt ²A^j^F_t^X^;1^_^F_Yt^;2) =P(Xt^;1A) : (2:10) Proof: See Nummelin (1984, p. 61 and Th.4.2, p. 62).

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8 2 MARKOV THEORY The compound chain is-irreducible, aperiodic and Harris recurrent (cf. Nummelin,^e 1984, Ch. 4.4).

The distribution of ^f(XtYt) t 0^g given by (2.8) can be written IP^e(X⁰ ²dx⁰Y⁰ =y⁰X¹ ²dx¹Y¹ =y¹:::Xk ²dxkYk =yk:::)

=^e(dx⁰^fy⁰^g)y⁰(x⁰dx¹)y¹(x¹) y_k^;1(xk^;1dxk)y_k(xk): (2:11) We simplify the notation and write IP to denote this distribution. If = x we write IPx which is the conditional distribution of (Y⁰^f(XtYt)t 1^g) given that X⁰ =x.

If the initial distribution is equal to (xy), i.e. Y⁰ = 1 X⁰ =x arbitrary, then IP(X¹ ²dx¹Y¹ =y¹:::Xk ²dxkYk =yk:::)

=(dx¹)y¹(x¹) y¹(x¹dx²)y²(x²) yk^;1(xk^;1dxk)yk(xk)

= IP(X⁰ ²dx¹Y⁰ =y¹X¹ ²dx²Y¹ =y²:::Xk ²dxk⁺¹Yk =yk⁺¹:::):(2:12) Let be a non-negative measurable function dened on (E ^f01^g)¹. Then by (2.12)

IE

h(X¹Y¹ :::)ⁱ= IE

h(X⁰Y⁰ :::)ⁱ= IE() : (2:13)

2.3 The invariant measure

In a general null recurrent chain ^fXt^gno marginal distribution function exists that can be estimated nonparametrically. There is a generalization of the distribution function in the invariant measure, however.

Let = = min^fn 0: Yn = 1^g and S = min^fn 1: Yn = 1^g. Since

fS =n^g = ^f\ⁿj⁼¹^;1(Yj = 0) Yn = 1^g and ^f = n^g =^fS = n^g^\^fY⁰ = 0^g, it follows from (2.11) (cf. Nummelin 1984, p. 63) that

IPx( = n) = (P ^;s)ⁿs(x) n 0

IP(S=n) = (P ^;s)ⁿ^;1s n 1 : (2:14) Dene s by

s(A) = s1A = IE

h^XS

n⁼¹1A(Xn)ⁱ A²^E : (2:15) Then by (2.14), (2.8) and (2.11)

s(A) = ^X¹

n⁼¹IE

h1A(Xn)1(S n)ⁱ

= ^X¹

n⁼¹(P ^;s)ⁿ^;11A

= Gs1A (2:16)

where

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2.3 The invariant measure 9

Gs ^def= ^X¹

n⁼⁰(P ^;s)ⁿ : (2:17)

This means thats =Gs and by (2.14) s(s) = IP(S < ¹). Since the split chain is Harris recurrent,

Gss(x) = IPx( <¹)1: (2:18) Thuss(s) = 1. From (2.17) we get

Gs =I + GsP ^;Gss (2:19)

which implies thats=sP. Thus sis an invariant measure. The results stated below can be found in Nummelin (1984).

Remark 2.1

^If is another invariant measure then = (s)s(p. 73). The invariant measure s is equivalent to , s(C) < ¹ for all small sets C and it is -nite (Prp.

5.6., p. 72).

The chain is positive recurrent if and only if s1E < ¹ (p. 68). In the positive recurrent case ^def= s=s1E is the unique stationary probability measure for ^fXt^g. In the latter situation, when the initial distribution of X⁰ is given by , ^fXt^g will evolve as a strictly stationary process having as its marginal distribution.

Remark 2.2

It is seen from(2.15)that^fXt^gis positive recurrent if and only if IES <

1.

It is seen from Remark 2.1 that although the representation of s given by (2.16) does depend of the atom (s), the measure s itself is independent of and only depends on s through a constant.

We may extend to the compound chain by ês(dx^fy^g) ^def= s(dx)y(x), and we have =ê êP.ê

Suppose that the original chain has a proper atom . Let s = 1 and = P = P(). Then P(xdy) 1(x)P(xdy) = s(x)(dy). Hence (2.4) is satised and all the formulae in this sub-section are still true if we dene the auxiliary process ^fYt^gby Yt= 1(Xt). It is common to denote Gs by G in this case.

2.3.1 A more general split chain

If m⁰ > 1 in (2.3), then the m⁰-step chain ^fXtm⁰^g satises (2.4) with transition probability P^m⁰ and (s) is an atom for this chain. The corresponding denitions of Gs

and s are given by

Gm⁰s =^X¹

t⁼⁰(P^m⁰ ^;s)^t s =Gm⁰s : (2:20)

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10 2 MARKOV THEORY Butsis still the unique invariant measure for the original chain which satisess(s) = 1.

2.4 Notation for functions in several variables

It is necessary to extend the notation of the preceding sub-sections to functions of several variables. All integrals involved will be assumed to be well-dened.

Recall that for g ²^G¹, s(g) =^Rs(dx)g(x) and (cf. (2.15) and (2.17)) Gsg(x) =^Z Gs(xdy)g(y) = IEx

h^X

n⁼⁰g(Xn)ⁱ: We introduce a useful transformation from^Gr to ^G¹.

Denition 2.1

Letr 1, and letg ²^Gr. Forr = 1andr = 2we deneI^eg(xdy)(1) = P(xdy)g(x) and I^eg(xdy)(2) = P(xdy)g(xy), respectively. For r > 2 let

Ieg(xdy)(r) =^Z P(xdx²) P(xr^;1dy)g(xx²:::xr^;1y) (2:21) where the integration is with respect to x²:::xr^;1 and whenever the right hand side is well-dened. Furthermore, dene

eg =I^eg1: (2:22)

Since ^Gr^;1 ^Gr for g ² ^Gr^;1, when r 2 we can write Iêg(xdy)(r) =Iêg(xdy)(r^; 1)P. An interpretation ofg is given byê

IEx

hg(X⁰X¹:::Xr^;1)ⁱ=I^eg1(x) =g(x)^e (2:23) and

IEx

h^X

j⁼⁰g(XjXj⁺¹:::Xj⁺r^;1)ⁱ=Gsg(x)^e (2:24) which is easily veried (cf. (2.15) and (2.17)). The right hand sides of (2.23) and (2.24) can be seen as convenient and compact ways of writing the conditional expectations on the left hand side. In the following we omit r in I^eg(xdy)(r):

If g ² ^G = ^G¹ then I^eg = IgP and g = I^e gP1 = g. In order to reduce the notation further we extend s to ¹_r⁼¹^Gr by

sg ^def= sg =^e ^Z s(dx¹)P(x¹dx²) P(xr^;1dxr)g(x¹:::xr) g ²^Gr : (2:25) We also extend the L^p spaces generated bys,

Lpr(s)^def= ⁿg ²^Gr:^kg^kpps ^def= sIê^jg^j^p1<¹ô p²(0¹) r 1 : (2:26) If we dene êsr(dx¹:::dxr) = s(dx¹)P(x¹dx²) P(xr^;1dxr), then Lpr(s) = L^p(êsr).

All of the notation in this sub-section is trivially extended to ^G_dr.

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2.5 Regularity concepts 11

2.5 Regularity concepts

We wish to formulate the regularity conditions to be stated in Section 3 and 4 in terms of standard Markov chain concepts, and we therefore include the denition of a special function and the concept of a f-regular measure. Theorem 2.1 will be used repeatedly in Section 3.

Denition 2.2

Let f be a non-negative measurable function, dened on E, which is s-integrable. The kernels ^fVAA²^E⁺^g are dened by

VAf(x) = IEx

h^XSA

n⁼¹f(Xn)ⁱ

where SA = min^fn 1 Xn ² A^g. Let be a nite measure on Ê and g ² L¹_r(s). The measure is g-regular if Iê^jg^j1 is -integrable and VAIê^jg^j1 is nite for all A. The function g is special ifg ²Lr(s)^\L²r(s), supIê^jg^j1 is nite andsupVAIê^jg^j1is nite for all A. The set D ²Ê⁺ is a special set if 1D is a special function.

To indicate that the restrictionm⁰ = 1 in (2.3) can be relaxed we state the following theorem for a generalm⁰. In this paper it will only be used with m⁰ = 1, however.

Theorem 2.1

Nummelin 1984, Prp. 5.13, p. 80)] Assume that^fXt^g is aperiodic. Let g ²L¹r(s) andm⁰ =I +P + +P^m⁰^;1. A nite measure isg-regular if and only if Gm⁰sm⁰Iê^jg^j1 is nite. Assume that supIê^jg^j1 is nite and g ² Lr(s)^\L²r(s). Then g is a special function if and only if supGm⁰sm⁰Iê^jg^j1 is nite. In particular, for r = 1all small functions are special.

Remark 2.3

Nummelin (1984) only treats r = 1, but since Iê^jg^j1 is a non negative function in one variable, the extension of the theorem to r > 1 is trivial. However, the condition thatsupIê^jg^j1is nite is in general a weaker condition than sup^jg^j<¹ when r > 1. In particular, even if g satises that supIê^jg^j1 and s^jg^j are nite, higher order moments of ^jg^j with respect to s(x⁰)P(x⁰dx¹) P(xr^;2dxr^;1) may not be nite.

2.6 -null recurrence and tail behaviour of recurrence times

To carry asymptotic theory through we need a regularity condition for the tail behaviour of the distribution of the recurrence timeS. Since this condition is crucial for most of what we are doing, we introduce it in a rather general way and then specialize to the case when (2.4) holds.

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12 2 MARKOV THEORY A positive function L dened on a¹), where a 0, is slowly varying at innity (Bingham et. al. 1989, p. 6) if

xlim^"1L(x)

L(x) = 1 for all > 0 and for all x²a¹): (2:27)

Denition 2.3

The Markov chain^fXt^g is -null recurrent if there exists a small non- negative function h, an initial measure , a constant ² (01) and a slowly varying function Lh so that

IE

h^Xn

t⁼⁰h(Xt)ⁱ 1

;(1 + )nLh(n) (2:28)

as n ^!¹.

Remark 2.4

^IfLandL⁰ are two slowly varying functions at innity, then they are said to be equivalent if limx^"1L(x)=L⁰(x) = 1. In all of our application of slowly varying functions they are only unique up to equivalence. Hence, when (2.28) is true, without any loss of generality we assume that Lh is normalized (Bingham et. al. , 1989, pp 15, 24), i.e., the function xLh(x) is strictly increasing and continuous in the interval x⁰¹) for some x⁰.

Let G⁽ⁿ⁾ =^Xⁿ

t⁼⁰P^t: (2:29)

The left hand side of (2.28) can be written as G⁽ⁿ⁾h. We rst prove that for a xed parameter (2.28) is actually a global property shared by all non-negative special functions.

Lemma 2.2

Assume that ^fXt^g is -null recurrent and aperiodic. Let (s) ^{be a xed} atom. Then we can nd anLs so that for all special functions f the asymptotic relation (2.28) holds with Lf =s(f)Ls where s is dened by (2.20).

Proof: Let and h be given by Denition 2.3 and the atom (s) be xed. Let

Ls ^def= Ls^hh : (2:30)

Using a null recurrent ratio limit theorem (Nummelin, 1984, Cor. 7.2(i), p. 131) and (2.28),

G⁽ⁿ⁾h

G⁽ⁿ⁾s = ^s(h)(1 + o(1)) : (2:31) Using (2.28) again and the above expression, it follows that

G⁽ⁿ⁾s 1

;(1 + )nLs(n) : (2:32)

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2.6 -null recurrence and tail behaviour of recurrence times 13 Letf be a given special function. Then by (2.31) with f instead of h and by (2.32) it follows that

G⁽ⁿ⁾f 1

;(1 + )ns(f)Ls(n) : (2:33)

2

Remark 2.5

If the atom (s) ⁱⁿ (2.4) is changed to (s⁰⁰) then (cf. Remark 2.1 ) s⁰ = s(s^s⁰) L^s⁰ s(s⁰)Ls: (2:34)

The asymptotic expression (2.32) is closely connected to the Tauberian theorem (Feller, 1971, p. 447):

Let ^fdn n 0^g be any non-negative sequence and let d(r) = ^P¹n⁼⁰rⁿdn be nite when ^jr^j is less than one. Moreover, let L¹ be slowly varying and ²0¹). Then

n

X

k⁼⁰dk 1

;(1 +)nL¹(n)⁽⁾d(r)(1^;r)^;L¹ 1 1^;r

(2:35)

when n^!¹ and r^"1^;, respectively. If ^fdn^g is monotone and > 0, then each of the conditions given by (2.35) is equivalent with

dn n^;1

;()L¹(n) : (2:36)

If (2.4) is true, the Tauberian theorem can be used to show that then the concept of -null recurrence implies a regularity condition for the tail behaviour of the distribution of the recurrence timeS.

Theorem 2.2

^Assume (2.4) is true. Then ^fXt^g is -null recurrent if and only if IP(S> n) = 1

;(1^; )nLs(n)

1 +o(1) : (2:37)

Remark 2.6

If (2.37) is true, then it is not dicult to show that sup^fp 0: IES_p<^1g= :

Thus, even thoughIES =¹ for a null recurrent process, if (2.4)and(2.37) hold, then IESp is nite for p small enough. For an ordinary random walk = 1=2 (^Kallianpur and Robbins, 1954) and hence IES_p < ¹ for 0 p < 1=2. Some other examples of

-null recurrent processes are given in Myklebust et al (1998a).

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14 2 MARKOV THEORY Proof of Theorem 2.2: Let G(r) = ^P¹k⁼⁰r^kP^k and Gs(r) =^P¹k⁼⁰r^k(P ^;s)^k. Then by (2.32) and (2.35) (with = and L¹ = Ls), -null recurrence is equivalent

with G(r)s(1^;r)^;Ls

1 1^;r

: (2:38)

We have Bn^def= IP(S > n) = (P ^;s)ⁿ1. If (2.37) holds, then by (2.35) and (2.36) (with = 1^; and L¹ =L⁰ = 1=Ls)

B(r)^def= ^X¹

k⁼⁰r^kBk =Gs(r)1 (1^;r)^;1L⁰ 1 1^;r

: (2:39)

Let bn = IP(S =n), wn = IP(Yn = 1) for n 1 and b⁰ = 0, w⁰ ^def= 1 and dene the corresponding generating functions w(r) and b(r). By a rst entrance decomposition

wn= IP(Yn = 1S n) +ⁿ^X^;1

k⁼¹IP(Yn^;k = 1)IP(S =k)

= ^Xⁿ

k⁼⁰wn^;kbk n 1 (2:40)

which shows that^fwn^gis an undelayed renewal sequence corresponding to the increment sequence ^fbn^g. By (2.40) we get w(r) = 1 + w(r)b(r). Since bn = Bn^;1 ^;Bn when n 1, we nd that b(r) = 1^;B(r)(1^;r). Hence

w(r) = 1

B(r)(1^;r) : (2:41)

By (2.8) and (2.13) we nd that wn = IP(Yn = 1) = IE

hs(Xn^;1)ⁱ = Pⁿ^;1s when n 1. This gives

w(r) = 1 + r^hG(r)sⁱ: (2:42)

Combining (2.41) and (2.42) we nally obtain 1 +r^hG(r)sⁱ= 1

Gs(r)1](1^;r) :

This identity in conjunction with (2.38) and (2.39) show the equivalence. ²

2.7 Weak limits for the number of regenerations

We assume aperiodicity and that (2.4) and (2.37) hold in this section. We also assume that Ls is normalized in the -null case which implies that the function

u(z)^def= zLs(z) z ²R⁺ (2:43) is strictly increasing in the interval z⁰¹) for somez⁰. Let

v(z) = u^(;1)(z) = inf^fs: u(s) > z^g: Then v(u(z)) = u(v(z)) = z for all z ²z⁰¹).

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2.7 Weak limits for the number of regenerations 15 A weak limit result can be derived from (2.37). Let T(n) denote the complete number of regenerations in the time interval 0n], i.e.,

T(n) = max^fk: k n^g if ⁰ n

0 otherwise, (2:44)

where

k =

(inf^fn 0:Yn = 1^g k = 0

inf^fn > k^;1:Yn= 1^g when k 1. (2:45) Then IE

hT(n)ⁱ= IE(^P_nj⁼¹Yj) =^Pⁿ_k^;1⁼⁰P^ks, and it follows by (2.28) (with h = s), IE

"

T(n)u(n)

#

= 1

;(1 + ) + o(1) : (2:46)

In some respectsT(n) corresponds to the total number of observations for a positive recurrent process, and it plays a crucial role in the asymptotics of Section 3 and 4. Our next task is to derive a functional limit theorem for T(n).

Let ^D0¹) denote the space of right continuous real valued functions with nite left hand limits, i.e. this is the space of cadlag functions dened on 0¹) (cf. Appendix B). We write ^L^{D 0}^;^!¹⁾ for weak convergence in^D0¹) and fd^;^! for convergence of nite dimensional laws. A Levy process is a stochastic process with stationary independent increments and sample paths in^D0¹). Consider the process

Sz(t)^def= 1v(z)

zt^]

X

k⁼¹(k^;k^;1) t²0¹) z ²R⁺ : (2:47) where zt] is the integer value of zt, i.e. the largest integer not exceeding zt.

By (2.37) (cf. Bingham et. al., 1989, p. 349) it follows that Sz ;fd!

z S : (2:48)

where S is the one-sided stable Levy process dened by the marginal characteristic function Eexp^f^{i S}⁽^t^)g] = exp^fⁱ^t^g for ²R and t² 0¹). Moreover, (cf. Kasahara, 1984)

Sz ^L^{D 0}^;^!z¹⁾ S : (2:49) The Mittag Leer process (cf. Kasahara, 1984) with parameter , M =

fM(t) t 0^g is dened as the inverse of S. It is a strictly increasing continuous stochastic process, and the characteristic functions describing the marginal distributions are given by

IE^hexp^f^{i M}⁽^t^)gⁱ= ^X¹

k⁼⁰

(it)^k

;(1 +k ) ²R t 0 : (2:50)

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16 2 MARKOV THEORY An alternative description is given by

E(M^m(1)) = m!

;(1 +m ) m 0 M(t)=^d tM(1): (2:51) We need the continuous extension of T(n). Let

Tn=ⁿT(nt])

u(n) t 0

o: (2:52)

The next theorem establishes a weak limit result, which will be of use in the asymptotic theory to be established in the sequel.

Theorem 2.3

^Let be any initial measure. Assume that (2.37) holds. Then IE

hT(n)ⁱ^m n^mL_ms(n)

;(1 +m ) (2:53)

and Tn^L^{D 0}^;n^!¹⁾ M : (2:54)

Proof: Let

T(n) =e ^Xⁿ

k⁼⁰Yk (2:55)

then T(n) = (T(n)^e ^;1)1(T(n) > 0) :^e (2:56)

The normalized continuous version is also denoted byT^en. It has the same properties as Tn. By Lemma A.3 in Appendix A we can write

IE

h

T(n)e ⁱ^m = ^X^m

k⁼¹

X

`²_mk

m

`

!

Jnk` (2:57)

where mk =^f` = (`¹:::`k)²^N⁺^k: ^P`i =m^g for k 1, and where Jnk`= ^Xⁿ

h¹⁼⁰ n^X^;h¹ h²⁼¹

n^;h¹^;X^;hk^;1 hk⁼¹ E

hY_h^`¹¹Y_h^`¹²⁺_h² Y_h^`¹^k⁺⁺_h_kⁱ

= ^Xⁿ

h¹⁼⁰ n^X^;h¹ h²⁼¹

n^;h¹^;^X^;hk^;1

h_k⁼¹ wh¹ wh_k wh = IP(Yh = 1) : (2:58) We can writeJnk` =Jnk since (2.58) shows that this quantity is independent of`. Let

Jk(r)^def= ^X¹

n⁼⁰Jnkrⁿ r ²01) : (2:59) Then it can be shown from (2.58), (2.39) and (2.41) that

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2.7 Weak limits for the number of regenerations 17

1

X

n⁼⁰Jnkrⁿ = ^X¹

n⁼⁰ n

X

h¹⁼⁰ n^X^;h¹ h²⁼¹

n^;h¹^;X^;h_k^;1

hk⁼¹ wh¹ wh_krⁿ

= ^X¹

h¹⁼⁰

1

X

n⁼h¹ nX^;h¹ h²⁼¹

n^;h¹^;X^;hk^;1

hk⁼¹ wh¹ whkrⁿ

= ^X¹

h¹⁼⁰

1

X

h²⁼¹

1

X

h_k⁼¹wh¹ wh_k

1

X

n⁼¹rⁿ⁺^h¹⁺^h^k

= w(r)(w(r)^;1)^k^;1r(1^;r)^;1

= w^k(r)(1^;r)^;1(1 +o(1)) and hence

Jk(r) w^k(r)(1^;r)^;1 (1^;r)^;^k^;1Lks 1 1^;r

(2:60)

as r ^" 1^;. From (2.35), (2.36) with = k + 1, L¹ = Lks, since ^fJnk n 1^g is a monotone sequence in n, (2.60) implies

Jnk n^kLks(n)

;(1 +k ) = u^k(n)

;(1 +k ) (2:61)

as n ^!¹. Inserting (2.61) into (2.57) gives IE

h

T(n)e ⁱ^m ^X^m

k⁼¹

(

X

`²_mk

m

`

!)n^kL_ks(n)

;(1 +k ) : (2:62)

and since mm =^f1^g =^f(1:::1)^g and ^m¹ =m! (cf Lemma A.3, Appendix A), we nally get by (2.62) that

IE

"

T(n)e

u(n)

#m

;(1 +m!m ) (2:63)

and (2.53) is proved.

We turn to the proof of (2.54). By the method of moments and (2.50) we nd that for each t

Ten(t)^;^!_n^d M(t) :

However, it is dicult to establish a functional weak convergence from the marginal convergences sinceM is not a Levy process. In order to prove (2.54) it is an advantage to use a continuous index i.e., Tz(t) ^def= T(zt])=u(z). By (2.48) and the proof of Theorem B.1 of Appendix B withS =A in that proof,

S_z^(;1) ^L^{D 0}^;^!_z¹⁾ M where S_z^(;1)(t) = inf^fx: Sz(x) > t^g: (2:64) In the rest of the proof we omit the index and writeSz =Sz and Sz^(;1) =Sz^(;1).

To prove (2.54) it is sucient to prove that sup

0<tK^jTz(t)^;Sz^(;1)(t)^j=oP(1) (2:65)

Nonparametric Estimation in Null Recurrent Time Series