Mixing*
It will be shown that, under mild conditions, GARCH processes are geometrically ergodic and β-mixing. These properties entail the existence of laws of large numbers and of central limit theorems (see Appendix A), and thus play an important role in the statistical analysis of GARCH processes. This chapter relies on the Markov chain techniques set out, for example, by Meyn and Tweedie (1996).
3.1 Markov Chains with Continuous State Space
Recall that for a Markov chain only the most recent past is of use in obtaining the conditional distribution. More precisely, (Xt) is said to be a homogeneous Markov chain, evolving on a space E (called the state space) equipped with a σ-field ε, if for all x E, and for all ε,
In this equation, Pt(x, B) corresponds to the transition probability of moving from the state x to the set in t steps. The Markov property refers to the fact that Pt(x, B) does not depend on Xr, r < s. The fact that this probability does not depend on s is referred to time homogeneity. For simplicity we write P(x, B) = P1(x, B). The function P : E × ε → [0, 1] is called a transition kernel and satisfies:
(i) ε, the function P(·, ) is measurable;
(ii) x E, the function P(x, ·) is a probability measure on (E,ε).
The law of the process (Xt) is characterized by an initial probability measure μ and a transition kernel P. For all integers t and all (t + l)-tuples (0,…, t) of elements of ε, we set
In what follows, (Xt) denotes a Markov chain on E = d and ε is the Borel σ -field.
The Markov chain (Xt) is said to be ø-irreducible for a nontrivial (that is, not identically equal to zero) measure ø on (E,ε), if
If (Xt) is ø-irreducible, it can be shown that there exists a maximal irreducibility measure, that is, an irreducibility measure M such that all the other irreducibility measures are absolutely continuous with respect to M. If M() = 0 then the set of points from which is accessible is also of zero measure (see Meyn and Tweedie, 1996, Proposition 4.2.2). Such a measure M is not unique, but the set
does not depend on the maximal irreducibility measure M. For a particular model, finding a measure that makes the chain irreducible may be a non trivial problem (but see Exercise 3.1 for an example of a time series model for which the determination of such a measure is very simple).
A ø-irreducible chain is called recurrent if
and is called transient if
Note that can be interpreted as the average time that the chain spends in when it starts at x. It can be shown that a ø-ireducible chain (Xt) is either recurrent or transient (see Meyn and Tweedie, 1996, Theorem 8.3.4). It is said that (Xt) is positive recurrent if
If a ø-irreducible chain is not positive recurrent, it is called null recurrent. For a ø-irreducible chain, positive recurrence is equivalent to the existence of a (unique) invariant probability measure (see Meyn and Tweedie, 1996, Theorem 18.2.2), that is, a probability π such that
An important consequence of this equivalence is that, for Markov time series, the issue of finding strict stationarity conditions reduces to that of finding conditions for positive recurrence. Indeed, it can be shown (see Exercise 3.2) that for any chain (Xt) with initial measure μ,
For this reason, the invariant probability is also called the stationary probability.
For a ø-irreducible chain, there exists a class of sets enjoying properties that are similar to those of the elementary states of a finite state space Markov chain. A set C ε is called a small set1 if there exist an integer m ≥ 1 and a nontrivial measure υ on ε such that
In the AR(1) case, for instance, it is easy to find small sets (see Exercise 3.4). For more sophisti-cated models, the definition is not sufficient and more explicit criteria are needed. For the so-called Feller chains, we will see below that it is very easy to find small sets. For a general chain, we have the following criterion (see Nummelin, 1984, Proposition 2.11): C ε+ is a small set if there exists ε+ such that, for all A, ε+, there exists T > 0 such that
If the chain is ø-irreducible, it can be shown that there exists a countable cover of E by small sets. Moreover, each set ε+ contains a small set C ε+. The existence of small sets allows us to define cycles for ø-irreducible Markov chains with general state space, as in the case of countable space chains. More precisely, the period is the greatest common divisor (gcd) of the set
where C ε+ is any small set (the gcd is independent of the choice of C). When d = 1, the chain is said to be aperiodic. Moreover, it can be shown (see Meyn and Tweedie, 1996, Theorem 5.4.4.) that there exist disjoint sets D1,…, Dd ε such that (with the convention Dd+1 = D1):
(i) i = l,…, d, x Di, P(x, Di+1) = 1;
(ii) ø(E-∪Di) = 0.
A necessary and sufficient condition for the aperiodicity of (Xt) is that there exists A ε+ such that for all A, ε+, there exists t > 0 such that
(see Chan, 1990, Proposition A1.2).
Geometric Ergodicity and Mixing
In this section, we study the convergence of the probability μ(Xt ·) to a probability π(·) independent of the initial probability μ, as t → ∞.
It is easy to see that if there exists a probability measure π such that, for an initial measure μ,
where μ(Xt ) is defined in (3.2) (for (0,…, t) = (E,…, E, )), then the probability π is invariant (see Exercise 3.3). Note also that (3.5) holds for any measure μ if and only if
On the other hand, if the chain is irreducible, aperiodic and admits an invariant probability π, for π -almost all x E,
where · denotes the total variation norm2 (see Meyn and Tweedie, 1996, Theorem 14.0.1). A chain (Xt) such that the convergence (3.6) holds for all x is said to be ergodic. However, this convergence is not sufficient for mixing. We will define a stronger notion of ergodicity.
The chain (Xt) is called geometrically ergodic if there exists ρ (0, 1) such that
Geometric ergodicity entails the so-called α- and β -mixing. The general definition of the α- and β-mixing coefficients is given in Appendix A.3.1. For a stationary Markov process, the definition of the α-mixing coefficient reduces to
where the first supremum is taken over the set of the measurable functions f and g such that | f | ≤ 1, | g | ≤ 1 (see Bradley, 1986, 2005). A general process X = (Xt) is said to be α-mixing (β-mixing) if αX(k) (βX(k)) converges to 0 as k → ∞. Intuitively, these mixing properties characterize the decrease in dependence when past and future become sufficiently far apart. The α-mixing is sometimes called strong mixing, but β-mixing entails strong mixing because αX(k) ≤ βX(k) (see Appendix A.3.1).
Davydov (1973) showed that for an ergodic Markov chain (Xt), of invariant probability measure π,
It follows that αX(k) = O(ρk) if the convergence (3.7) holds. Thus
Two Ergodicity Criteria
For particular models, it is generally not easy to directly verify the properties of recurrence, existence of an invariant probability law, and geometric ergodicity. Fortunately, there exist simple criteria on the transition kernel.
We begin by defining the notion of Feller chain. The Markov chain (Xt) is said to be a Feller chain if, for all bounded continuous functions g defined on E, the function of x defined by E(g(Xt)|Xt−1 = x) is continuous. For instance, for an AR(1) we have, with obvious notation,
The continuity of the function x → g(θx + y) for all y, and its boundedness, ensure, by the Lebesgue dominated convergence theorem, that (Xt) is a Feller chain. For a Feller chain, the compact sets C ε+ are small sets (see Feigin and Tweedie, 1985).
The following theorem provides an effective way to show the geometric ergodicity (and thus the β-mixing) of numerous Markov processes.
Theorem 3.1 (Feigin and Tweedie (1985, Theorem 1)) Assume that:
(i) (Xt) is a Feller chain;
(ii) (Xt) is ø-irreducible;
(iii) there exist a compact set A E such that ø (A) > 0 and a continuous function V : E → + satisfying
and for δ > 0,
Then (Xt) is geometrically ergodic.
This theorem will be applied to GARCH processes in the next section (see also Exercise 3.5 for a bilinear example). In equation (3.10), V can be interpreted as an energy function. When the chain is outside the center A of the state space, the energy dissipates, on average. When the chain lies inside A, the energy is bounded, by the compactness of A and the continuity of V. Sometimes V is called a test function and (III) is said to be a drift criterion.
Let us explain why these assumptions imply the existence of an invariant probability measure. For simplicity, assume that the test function V takes its values in [l,+∞), which will be the case for the applications to GARCH models we will present in the next section. Denote by P the operator which, to a measurable function f in E, associates the function Pf defined by
Let Pt be the tth iteration of P, obtained by replacing P(x, dy) by Pt (x, dy) in the previous integral. By convention P0f = f and P0(x, A) = A. Equations (3.9) and (3.10) and the boundedness of V by some M > 0 on A yield an inequality of the form
where b = M − (1 − δ)., Iterating this relation t times, we obtain, for x0 A
It follows (see Exercise 3.6) that there exists a constant k > 0 such that for n large enough,
The sequence Qn(x0, ·) being a sequence of probabilities on (E, ε), it admits an accumulation point for vague convergence: there exist a measure π of mass less than 1 and a subsequence (nk) such that for all continuous functions f with compact support,
In particular, if we take f = A in this equality, we obtain π(A) ≥ k, thus π is not equal to zero. Finally, it can be shown that π is a probability and that (3.13) entails that π is an invariant probability for the chain (Xt) (see Exercise 3.7).
For some models, the drift criterion (iii) is too restrictive because it relies on transitions in only one step. The following criterion, adapted from Meyn and Tweedie (1996, Theorems 19.1.3, 6.2.9 and 6.2.5), is an interesting alternative relying on the transitions in n steps.
Theorem 3.2 (Geometric ergodicity criterion) Assume that:
(i) (Xt) is an aperiodic Feller chain;
(ii) (Xt) is ø-irreducible where the support of ø has nonempty interior;
(iii) there exist a compact C E, an integer n≥1 and a continuous function V : E → + satisfying
and for δ > 0 and b >0,
Then (Xt) is geometrically ergodic.
The compact C of condition (iii) can be replaced by a small set but the function V must be bounded on C. When (Xt) is not a Feller chain, a similar criterion exists, for which it is necessary to consider such small sets (see Meyn and Tweedie, 1996, Theorem 19.1.3).
3.2 Mixing Properties of GARCH Processes
We begin with the ARCH(l) process because this is the only case where the process (t) is Markovian.
The ARCH(l) Case
Consider the model
where ω > 0, α ≥ 0 and (ηt) is a sequence of iid (0, 1) variables. The following theorem establishes the mixing property of the ARCH(l) process under the necessary and sufficient strict stationarity condition (see Theorem 2.1 and (2.10)). An extra assumption on the distribution of ηt is required, but this assumption is mild:
Assumption A The law Pη of the process (ηt) is absolutely continuous, of density f with respect to the Lebesgue measure λ on (, ()). We assume that
and that there exists τ > 0 such that
Note that this assumption includes, in particular, the standard case where f is positive over a neighborhood of 0, possibly over all . We then have η0 = 0. Equality (3.17) implies some (local) symmetry of the law of (ηt). This symmetry facilitates the proof of the following theorem, but It can be omitted (see Exercise 3.8).
Theorem 3.3 (Mixing of the ARCH(l) model) Under Assumption A and for
the nonanticipative strictly stationary solution of the ARCH (1) model (3.16) is geometrically ergodic, and thus geometrically β-mixing.
Proof. Let ψ(x) = (ω + αx2)½. A process (t) satisfying
where ηt is independent of t-i, i > 0, is clearly a homogenous Markov chain on (, ()), with transition probabilities
We will show that the conditions of Theorem 3.1 are satisfied.
Step (i). We have
If g is continuous and bounded, the same is true for the function x → g{ψ(x)y], for all y. By the Lebesgue theorem, it follows that (t) is a Feller chain.
Step (ii). To show the ø-irreducibility of the chain, for some measure ø, assume for the moment that η0 = 0 in Assumption A. Suppose, for instance, that f is positive on [0, τ). Let ø be the restriction of the Lebesgue measure to the interval . Since Ψ(x) ≥ , It can be seen that
It follows that the chain (t) is ø-irreducible. In particular, ø = λ if ηt has a positive density over .
The proof of the irreducibility in the case η0>0 is more difficult. First note that
Now E log α by (3.18). Thus we have
Let τ′ (0, τ) be small enough such that
Iterating the model, we obtain that, for 0 = x fixed,
Is a diffeomorphism between open subsets of t. Moreover, in view of Assumption A, the vector Yt has a density on t. The same is thus true for Zt, and It follows that, given 0=x,
We now Introduce the event
Assumption A implies that (Ξt) > 0. Conditional on Ξt, we have
Since the bounds of the interval It are reached, the intermediate value theorem and (3.19) entail that, given 0 = x, has, conditionally on Ξt, a positive density on It. It follows that
where Jt = {x | x2 It}. Let
and let λJ be the restriction of the Lebesgue measure to J. We have
The chain (t) is thus ø-irreducible with ø = λJ.
Step (iii). We shall use Lemma 2.2. The variable α Is almost surely positive and satisfies E(α) = α < ∞ and Elogα < 0, in view of assumption (3.18). Thus, there exists s > 0 such that
where μ2s = Es. The proof of Lemma 2.2 shows that we can assume s ≤ 1. Let V(x) = 1 + x2s. Condition (3.9) is obviously satisfied for all x. Let 0 < δ < 1 − c and let the compact set
Since A is a nonempty closed interval with center 0, we have ø(A) > 0. Moreover, by the inequality (a+b)s≤as+bs for a, b ≥ 0 and s [0, 1] (see the proof of Corollary 2.3), we have, for x A,
which proves (3.10). It follows that the chain (t) is geometrically ergodic. Therefore, in view of (3.8), the chain obtained with the invariant law as Initial measure is geometrically β-mixing. The proof of the theorem is complete.
Remark 3.1 (Case where the law of ηt does not have a density) The condition on the density of ηt is not necessary for the mixing property. Suppose, for example, that , a.s. (that is, ηt takes the values − 1 and 1, with probability ½). The strict stationarity condition reduces to α < 1 and the strictly stationary solution is a.s This solution is mixing since it is an independent white noise.
Another pathological example is obtained when ηt has a mass at 0: (ηt = 0) = θ > 0. Regardless of the value of α, the process is strictly stationary because the right-hand side of inequality (3.18) is equal to +∞. A noticeable feature of this chain is the existence of regeneration times at which the past is forgotten. Indeed, if ηt = 0 then t = 0, , … It is easy to see that the process is then mixing, regardless of α.
The GARCH(1,1) Case
Let us consider the GARCH(1, 1) model
where ω>0, α ≥0, β ≥ 0 and the sequence (ηt) is as in the previous section. In this case (σt) is Markovian, but (t) is not Markovian when β > 0. The following result extends Theorem 3.3.
Theorem 3.4 (Mixing of the GARCH(1,1) model) Under Assumption A and if
then the nonanticipative strictly stationary solution of the GARCH(1, 1) model (3.22) is such that the Markov chain (σt) is geometrically ergodic and the process (t) is geometrically β-mixing.
Proof. If α = 0 the strictly stationary solution is iid, and the conclusion of the theorem follows in this case. We now assume that α > 0. We first show the conclusions of the theorem that concern the process (σt). A homogenous Markov chain (σt) is defined on (+,(+)) by setting, for t ≥ 1,
where a(x) = ax2 + β. Its transition probabilities are given by
where x = {η; {ω + a(η)x2}½ }. We show the stated results by checking the conditions of Theorem 3.1.
Step (i). The arguments given in the ARCH(l) case, with
are sufficient to show that (σt) is a Feller chain.
Step (ii). To show the irreducibility, note that (3.23) implies
since |ηt| ≥ η0 a.s. and a(·) is an increasing function. Let τ′ (0,τ) be small enough such that
If σ0 = x +, we have, for t > 0,
Conditionally on Ξt, defined in (3.20), we have
Let
Then, given σ0 = x,
σt has, conditionally on Ξt, a positive density on Jt
where Jt={x +|x2 Ixt}. Let λJ be the restriction of the Lebesgue measure to . We have
The chain (σt) is thus ø-irreducible with ø = λJ.
Step (iii). We again use Lemma 2.2. By the arguments used in the ARCH(l) case, there exists s [0, 1] such that
Define the test function by V(x) = 1 + x2s, let 0 ≤ δ ≤ 1 − c1 and let the compact set
We have, for x A,
which proves (3.10). Moreover (3.9) is satisfied.
To be able to apply Theorem 3.1, it remains to show that ø(A) >0 where ø is the above irreducibility measure. In view of the form of the intervals I and A, it is clear that, denoting by Å the interior of A,
Therefore, it suffices to choose δ sufficiently close to 1 − c1 so that the last inequality is satisfied. For such a choice of δ, the compact set A satisfies the assumptions of Theorem 3.1. Consequently, the chain (σt) is geometrically ergodic. Therefore the nonanticipative strictly stationary solution (σt), satisfying (3.24) for t , is geometrically β-mixing.
Step (iv). Finally, we show that the process (t) inherits the mixing properties of (σt). Since t = σtηt, it is sufficient to show that the process Yt = (σt, ηt)′ enjoys the mixing property. It is clear that (Yt) is a Markov chain on + × equipped with the Borel σ-field. Moreover, (Yt) is strictly stationary because, under condition (3.23), the strictly stationary solution (σt) is nonanticipative, thus Yt is a function of ηt, ηt−1,… Moreover, σt is independent of ηt. Thus the stationary law of (Yt) can be denoted by Y = σ η where σ denotes the law of σt and η that of ηt. Let t(y, ·) the transition probabilities of the chain (Yt). We have, for y = (y1, y2) + × , 1 (+), 2 () and t > 0,
It follows, since η is a probability, that
The right-hand side converges to 0 at exponential rate, in view of the geometric ergodicity of (σt). It follows that (Yt) is geometrically ergodic and thus β-mixing. The process (t) is also β-mixing, since t is a measurable function of Yt.
Theorem 3.4 is of interest because it provides a proof of strict stationarity which is completely different from that of Theorem 2.8. A slightly more restrictive assumption on the law of ηt has been required, but the result obtained in Theorem 3.4 is stronger.
The ARCH(q) Case
The approach developed in the case q = 1 does not extend trivially to the general case because (t) and (σt) lose their Markov property when p>1 or q > 1. Consider the model
where ω > 0, αi ≥ 0, i = 1.,…, q, and (ηt) is defined as in the previous section. We will once again use the Markov representation
where
Recall that γ denotes the top Lyapunov exponent of the sequence {At, t }.
Theorem 3.5 (Mixing of the ARCH(q) model) If ηt has a positive density on a neighborhood of 0 and γ < 0, then the nonanticipative strictly stationary solution of the ARCH(q) model (3.25) is geometrically β-mixing.
Proof. We begin by showing that the nonanticipative and strictly stationary solution () of the model (3.26) is mixing. We will use Theorem 3.2 because a one-step drift criterion is not sufficient.
Using (3.26) and the independence between ηt and the past of it can be seen that the process () is a Markov chain on (+)q equipped with the Borel σ-field, with transition probabilities
The Feller property of the chain () is obtained by the arguments employed in the ARCH(l) and GARCH(1, 1) cases, relying on the independence between ηt and the past of as well as on the continuity of the function → t + At.
In order to establish the irreducibility, let us consider the transitions in q steps. Starting from 0 = , after q transitions the chain reaches a state q of the form
where the functions ψi- are such that ψi(·) ≥ ω > 0. Let τ > 0 be such that the density f of ηi be positive on (−τ, τ), and let ø be the restriction to [0, ωτ2[q of the Lebesgue measure λ on q.
It follows that, for all = 1 × … × q ((+)q), ø()>0 implies that, for all y1,…,yq (+)qand for all i = 1,…, q,
which implies in turn that, for all (+)q, Pq(,) > 0. We conclude that the chain () is ø-irreducible.
The same argument shows that
The criterion given in (3.4) can then be checked, which implies that the chain is aperiodic.
We now show that condition (iii) of Theorem 3.2 is satisfied with the test function
where · denotes the norm A = Σ |Aij| of a matrix A = (Aij) and s (0, 1) is such that
for some integer k0 ≥ 1. The existence of s and k0 is guaranteed by Lemma 2.3. Iterating (3.26), we have
The norm being multiplicative, it follows that
The inequality comes from the independence between At and t−i for i > 0. The existence of the expectations on the right-hand side of the inequality comes from arguments used to show (2.33). Let δ > 0 such that 1. − δ > p and let C be the subset of (+)p+q defined by
We have C ≠ because K > 1. − δ. Moreover, C is compact because 1 − δ − ρ > 0. Condition (3.14) is clearly satisfied, V being greater than 1. Moreover, (3.15) also holds true for n = k0−1. We conclude that, in view of Theorem 3.2, the chain is geometrically ergodic and, when it is initialized with the stationary measure, the chain is stationary and β-mixing.
Consequently, the process (), where (tt) is the nonanticipative strictly stationary solution of model (3.25), is βmixing, as a measurable function of This argument is not sufficient to conclude concerning (t). For k > 0, let
where f and g are measurable functions. Note that
Similarly, we have E(Zk |, t ) = E(Zk|,t ≥ k) and we have independence between Y0 and Zk conditionally on (). Thus, we obtain
It follows, in view of the definition (A.5) of the strong mixing coefficients, that
In view of (A.6), we also have . Actually, (A.7) entails that the converse inequalities are always true, so we have and β2(k) = β(k). The theorem is thus shown.
A major reference on ergodicity and mixing of general Markov chains is Meyn and Tweedie (1996). For a less comprehensive presentation, see Chan (1990), Tj0stheim (1990) and Tweedie (1998). For survey papers on mixing conditions, see Bradley (1986, 2005). We also mention the book by Doukhan (1994) which proposes definitions and examples of other types of mixing, as well as numerous limit theorems.
For vectorial representations of the form (3.26), the Feller, aperiodicity and irreducibility properties were established by Cline and Pu (1998, Theorem 2.2), under assumptions on the error distribution and on the regularity of the transitions.
The geometric ergodicity and mixing properties of the GARCH(p, q) processes were established in the PhD thesis of Boussama (1998), using results of Mokkadem (1990) on polynomial processes. The proofs use concepts of algebraic geometry to determine a subspace of the states on which the chain is irreducible. For the GARCH(1, 1) and ARCH(q) models we did not need such sophisticated notions. The proofs given here are close to those given in Francq and Zakoïan (2006a), which considers more general GARCH(1, 1) models. Mixing properties were obtained by Carrasco and Chen (2002) for various GARCH-type models under stronger conditions than the strict stationarity (for example, α + β < 1 for a standard GARCH(1, 1); see their Table 1). Recently, Meitz and Saikkonen (2008a, 2008b) showed mixing properties under mild moment assumptions for a general class of first-order Markov models, and applied their results to the GARCH(1, 1).
The mixing properties of ARCH(∞) models are studied by Fryzlewicz and Subba Rao (2009). They develop a method for establishing geometric ergodicity which, contrary to the approach of this chapter, does not rely on the Markov chain theory. Other approaches, for instance developed by Ango Nze and Doukhan (2004) and Hormann (2008), aim to establish probability properties (different from mixing) of GARCH-type sequences, which can be used to establish central limit theorems.
3.1 (Irreducibility condition for an AR(1) process)
Given a sequence (q)t of iid centered variables of law P which is absolutely continuous with respect to the Lebesgue measure λ on , let (Xt)t be the AR(1) process defined by
where θ .
(a) Show that if P has a positive density over , then (Xt) constitutes a λ-irreducible chain.
(b) Show that if the density of t is not positive over all , the existence of an irreducibility measure is not guaranteed.
3.2 (Equivalence between stationarity and invariance of the initial measure) Show the equivalence (3.3).
3.3 (Invariance of the limit law)
Show that if π is a probability such that for all , μ(Xt ) → π () when t→∞, then π is invariant.
3.4 (Small sets for AR(1))
For the AR(1) model of Exercise 3.1, show directly that if the density f of the error term is positive everywhere, then the compacts of the form [−c, c], c> 0, are small sets.
3.5 (From Feigin and Tweedie, 1985)
For the bilinear model
where (t) is as in Exercise 3.1(a), show that if
then there exists a unique strictly stationary solution and this solution is geometrically ergodic.
3.6 (Lower bound for the empirical mean of the pt(x0, A))
Show inequality (3.12).
3.7 (Invariant probability)
Show the invariance of the probability π satisfying (3.13).
Hints: (i) For a function g which is continuous and positive (but not necessarily with compact support), this equality becomes
(see Meyn and Tweedie, 1996, Lemma D.5.5).
(ii) For all σ-finite measures μ on (, ()) we have
(see Meyn and Tweedie, 1996, Theorem D.3.2).
3.8 (Mixing of the ARCH(1) model for an asymmetric density)
Show that Theorem 3.3 remains true when Assumption A is replaced by the following: The law Pη is absolutely continuous, with density f, with respect to λ. There exists τ > 0 such that
where and .
3.9 (A result on decreasing sequences)
Show that if uη is a decreasing sequence of positive real numbers such that we have supnnun<∞. Show that this result applies to the proof of Corollary A.3 in Appendix A.
3.10 (Complements to the proof of Corollary A.3)
Complete the proof of Corollary A.3 by showing that the term d4 is uniformly bounded in t, h and k.
3.11 (Nonmixing chain)
Consider the nonmixing Markov chain defined in Example A.3. Which of the assumptions (i)–(iii) in Theorem 3.1 does the chain satisfy and which does it not satisfy?
1 Meyn and Tweedie (1996) introduce a more general notion, called a ‘petite set’, obtained by replacing, in the definition, the transition probability in m steps by an average of the transition probabilities, , where (am) is a probability distribution.
2 The total variation norm of a (signed) measure m is defined by m = sup ∫ fdm, where the supremum is taken over {f : E → , f measurable and | f | ≤ 1}.