V -geometrical ergodicity of Markov kernels via finite-rank approximations

Under the standard drift/minorization and strong aperiodicity assumptions, this paper provides an original and quite direct approach of the V -geometrical ergodicity of a general Markov kernel P , which is by now a classical framework in Markov modelling. This is based on an explicit approximation of the iterates of P by positive finite-rank operators, combined with the Krein-Rutman theorem in its version on topological dual spaces. Moreover this allows us to get a new bound on the spectral gap of the transition kernel. This new approach is expected to shed new light on the role and on the interest of the above mentioned drift/minorization and strong aperiodicity assumptions in V -geometrical ergodicity. AMS subject classification : 60J05


Introduction
Throughout the paper P is a Markov kernel on a measurable space (X, X ). For any positive measure µ on X and any µ-integrable function f : X → C, µ(f ) denotes the integral f dµ. When P admits a unique invariant distribution denoted by π, an important question in the theory of Markov chains is to find condition for the n−th iterate P n of P to converge to π when n → +∞, and to control P n − π(·)1 X for some functional norm. In this paper we consider the standard V −weighted norm · V associated with some [1, +∞)-valued function V on X. Then the property P n − π(·)1 X V := sup implies that there exists ρ ∈ (0, 1) such that P n − π(·)1 X V = O(ρ n ): this corresponds to the so-called V -geometrical ergodicity property, see [MT93,RR04,KM03]. The infimum of all the real numbers ρ such that the previous property holds true is the so-called spectral gap of P , denoted by ρ V (P ).
Since the classical work by Meyn and Tweedie [MT93,MT94], it is well known that P is V -geometrically ergodic provided that usual irreducibility/aperiodicity assumptions hold true and that the following drift/minorization conditions are fulfilled: there exist S ∈ X , called a small set, and a positive measure ν on (X, X ) such that ∃δ ∈ (0, 1), ∃L > 0, P V ≤ δ V + L 1 S , ∀x ∈ X, ∀A ∈ X , P (x, A) ≥ ν(1 A ) 1 S (x).
Condition (M) when the small set S is the entire state space X is the so-called Doeblin condition. The proofs in [MT93,MT94,Bax05] are based on renewal theory involving the study of the return times to the small set S and Kendall's theorem. Actually the renewal theory applies to the atomic case (i.e. when S is an atom), and it has to be applied to the split chain in the general case.
In this paper, under Assumptions (D)-(M) and the (strong) aperiodicity condition as in [Bax05] ν(1 S ) > 0, we revisit the V -geometrical ergodicity property of P thanks to a simple constructive approach based on an explicit approximation of the iterates of P by positive finite-rank operators, combined with Krein-Rutman theorem [KR50]. This theorem can be thought of as an abstract dual Perron-Frobenius statement. It is stated at the end of this section in our specific case of positive operators acting on a weighted-suppremum norm space.
In Section 3, under Conditions (D)-(M), the unique invariant distribution π of P is obtained from the explicit series which extends a well-known formula when P satisfies the Doeblin condition, see [LC14], or when P is irreducible and recurrent positive according to [Num84,p 74]. More important, as a result of the above assertion (ii), we easily derive the rate β n (V ) = O(γ n ) as well as an approximation of π by an explicit sequence of probability measures with the same convergence rate. In Sections 4 and 5, under the additional assumption (SA), an original proof of the V -geometrical ergodicity is derived from the results of Sections 2-3. More precisely, setting 1 n the V -geometrical ergodicity follows from the following bounds of the spectral gap of P ρ V (P ) ≤ max r, S ≤ min |z| : 1 < |z| < 1/r, with the convention that the above minimum equals to 1/r if the related set is empty (in this case ρ V (P ) ≤ r ).
Although the results of Section 5 seem to be close to those in [Bax05], where a real number similar to S is also introduced, it is worth noticing that they differ completely from their content and their proofs. Indeed, on the one hand the renewal theory is not used here, on the other hand no intermediate Markov kernel is required in our work, in particular we do not use the split chain. Our method is mainly based on the Krein-Rutman theorem. Recall that the classical Perron-Frobenius theorem is a useful result for obtaining positive eigenvectors belonging to the maximal positive eigenvalue of a finite non-negative matrix. Here the Krein-Rutman theorem plays the same role (on the dual side). The following four stages outline our approach. First, the minorization condition (M) provides the positive finite-rank operator T n in the above assertion (i). Let us mention that such an approach has been used in [HL19] to study inhomogeneous products of Markov kernels satisfying the Doeblin condition. Second, the geometric rate of (P − T ) n V is obtained under Conditions (D)-(M) thanks to the Krein-Rutman theorem. Third, the existence and uniqueness of the invariant distribution π is deduced from the Krein-Rutman theorem too. Four, standard arguments on power series are used to prove Inequalities (2) under the additional assumption (SA).
As mentioned in [Bax05] (see also the references therein), the bounds of ρ V (P ) obtained in the literature may be still quite far off ρ V (P ), and we do not presume to give here a better bound of ρ V (P ). Actually this new approach is expected to shed new light, as for instance in [HM11], on the role of Assumptions (D)-(M)-(SA) in the study of the V -geometrical ergodicity.

Notations and basic material
Let V : X →[1, +∞) be a measurable function such that V (x 0 ) = 1 for some x 0 ∈ X. Let (B V , · V ) denote the weighted-supremum Banach space If Q is a bounded linear operator on B V , its operator norm Q V is defined by If Q 1 and Q 2 are bounded linear operators on B V , we write Q 1 ≤ Q 2 when the following property holds: ∀f ∈ B V , f ≥ 0, Q 1 f ≤ Q 2 f . Under Assumption (D), the following functional action of P is well-defined and provides a bounded linear operator on B V . Recall that P is said to be V -geometrically ergodic if there exists a P -invariant probability measure π on (X, X ) such that π(V ) < ∞ and if there exist some rate ρ ∈ (0, 1) and constant C ρ > 0 such that Denoting by Π the rank-one operator f → π(f )1 X on B V , Property (3) rewrites as The spectral gap of P , denoted by ρ V (P ), is defined as the spectral radius r(P − Π) of the operator P − Π, that is Equivalently ρ V (P ) is the infimum of all the real numbers ρ such that (3) holds true for some positive constant C ρ . Finally B V denotes the topological dual space of B V , that is the Banach space composed of all the continuous linear forms on B V , equipped with its usual norm: Finally, for the sake of simplicity, let us state the Krein-Rutman theorem for the positive operators on B V . In such a context, a proof can be directly obtained from [MN91, Th 4.1.5, p 251] using E := B V and · e := · V .
Krein-Rutman theorem If L is a positive bounded linear operator on B V such that its spectral radius r(L) = lim n L n 1/n V > 0, then there exists a non-trivial non-negative η ∈ B V such that η • L = r(L) η.
2 Approximation of P n by a positive finite-rank operator Let P be a Markov kernel satisfying Conditions (D)-(M). We set β 1 (·) := ν(·), and for every n ≥ 2, the element β n (·) of B V is defined by the following recursive formula : Note that β 1 (·) = ν(·) is defined as a positive measure on (X, X ) and that β 1 (V ) = ν(V ) < ∞ from (D)-(M). Thus β 1 (·) defines a non-negative element of B V . It follows from induction that, for every n ≥ 1, β n (·) is well defined as an element of B V . Actually the next proposition shows that, for every n ≥ 1, β n (·) can be defined as a positive measure on (X, X ) such that β n (V ) < ∞. Let T be the rank-one non-negative operator on B V defined by : It follows from (M) that 0 ≤ T ≤ P .
Proposition 2.1 Assume that P satisfies Assumptions (D)-(M). Then Moreover, for every n ≥ 1, β n is a positive measure on (X, X) such that β n (V ) < ∞, that is: there exists a positive measure on (X, X) (still denoted by β n ) such that X V dβ n < ∞ and: ∀f ∈ B V , β n (f ) = X f dβ n . Proof. The first equality in (7) is just the definition of T n . That 0 ≤ T n ≤ P n follows from 0 ≤ T ≤ P . The second equality in (7) for n = 1 is obvious from the definition of T . Now assume that this second equality holds true for some n ≥ 1. Then (6). This provides the second equality in (7) by induction.
As already mentioned β 1 (·) = ν(·) is defined as a positive measure on (X, X ) such that β 1 (V ) < ∞. Next, for every n ≥ 1, the element β n (·) is defined as an element of B V and for every f ∈ B V , we have from (6) and then from (7) It follows that β n (·) is a non-negative element of B V since P n−1 ≥ T n−1 . To complete the proof, let us prove by induction that, for every n ≥ 1, β n is a positive measure on (X, X) such that β n (V ) < ∞. Assume that, for some n ≥ 2, the following property holds: for every That is: for every 1 ≤ k ≤ n − 1 there exists a positive measure on (X, X) (still denoted by β k ) such that (6) is a finite linear combination of positive measures on (X, X). It follows that β n (·) is itself a positive measure on (X, X) since we have proved that β n is non-negative.
Here a short proof of (11) is given to highlight the use of the Krein-Rutman theorem.
Note that, for every n ≥ 1, the operator T n defined in Proposition 2.1 is positive and finiterank, more precisely Im(T n ) is contained in the n−dimensional subspace of B V generated by the functions 1 S , P 1 S , . . . , P n−1 1 S . The following corollary is a direct consequence of Proposition 2.1 and Theorem 2.1.
Corollary 2.1 Assume that P satisfies (D)-(M). Then, for every γ ∈ (r, 1), there exists C γ > 0 such that Under Conditions (D)-(M), Inequality (12) provides a geometric convergence rate for the difference between the n-th iterate of P and the positive finite-rank operator T n . This will be a central preliminary property for obtaining the results of Sections 3, 4 and 5.
3 Existence and approximation of π with the β k 's defined in (6). It follows from Proposition 2.1 that µ n is a positive measure on (X, X ) such that µ n (V ) < ∞. We provide a very short proof that P has a unique invariant probability π with a simple representation from the β k 's.
Proof. Under Condition (D), we know from the proof of Theorem 2.1 that the spectral radius r(P ) of P is one. Next, we know from the Krein-Rutman theorem that there exists a non-zero and non-negative element φ ∈ B V such that φ • P = φ. We may assume that φ V = 1. We obtain using the P -invariance of φ and (12) that for every γ ∈ (r, 1) where C γ is given in Corollary 2.1. It follows that φ = φ(1 S ) +∞ k=1 β k in B V . Actually this series absolutely converges in B V since we have for every n ≥ 2 β n V ≤ ν V P n−1 − T n−1 V ≤ ν(V )C γ γ n−1 from (9) and Corollary 2.1, and from ν V = ν(V ). Next µ := +∞ k=1 β k defines a sigmaadditive positive measure from Proposition 2.1. Since β k (1 X ) ≤ β k (V ) = β k V for any k ≥ 1, we have µ(1 X ) ≤ µ(V ) < +∞ and µ is a finite positive measure. Thus φ is a finite positive measure and φ is a P -invariant probability up to a normalization factor.
The following theorem states that the P -invariant probability π may be approximated by a sequence of probability measures defined from the β k 's. Indeed, µ n (1 X ) ≥ β 1 (1 X ) = ν(1 X ) > 0 for every n ≥ 1. Thus, we can define from (13) the following probability measure µ n (·) on (X, X ) such that µ n (V ) < ∞: Theorem 3.2 Assume that P satisfies (D)-(M). Let γ be such that γ ∈ (r, 1), and let n 0 be the smallest integer number such that L C γ γ n 0 + δ < 1, with C γ given in Corollary 2.1. Then the following assertion holds for the P -invariant probability π: Proof. We have from (17) and from the triangle inequality that Using the notations of Proposition 2.1 and Corollary 2.1, we deduce from the P −invariance of π that π • T n = π(1 S )µ n . It follows from (12) that We deduce from (D) that π(V ) ≤ δ π(V )+L π(1 S ). Thus π(1 S ) > 0 since δ < 1 and π(V ) > 0 and we have Therefore, we obtain from (20) Let us control the second term in the right hand side of Inequality (19). Property (22) with f := 1 X gives Let n ≥ n 0 . We know from (23) that π(1 S )µ n (1 (1 − δ) > 0 from π(1 S ) ≤ 1 and the definition of n 0 . It follows from (23) and from the last inequality that Next, it remains to note that (14) and (21) The proof of Inequality (18) is complete.
Remark 3.1 Theorem 3.1 asserts the existence of a unique invariant probability when X is a general state space and P satisfies the conditions (D)-(M). Under topological assumptions on X such a statement can be simply obtained by using Prohorov's theorem. This is the case when P satisfies the drift condition (D) provided that X is a separable complete metric space and that V has compact level sets (for completenes a proof is postponed to Proposition A.1).
Remark 3.2 It follows from (9) and (7) that β k = ν(P − T ) k−1 for every k ≥ 1, so that the series representation (14) of π reduces to π = π(1 S ) ν +∞ k=0 (P −T ) k . Such a representation is well known when P satisfies the Doeblin condition (i.e. X is a small set, e.g. see [LC14]) and when P is irreducible and recurrent positive by using the renewal theory, see [Num84,p. 74]. Note that Theorem 3.1 gives this formula with the additional geometric rate (15) which is central for analysing the power series introduced in the next section.

Some relevant power series
In this short section some power series related to the β k (·)'s are introduced and we prove a result that highlights the interest of Property (15) and the role of Assumption (SA). For every τ > 0 we set D(0, τ ) := {z ∈ C : |z| < τ } and D(0, τ ) := {z ∈ C : |z| ≤ τ }. β k (f ) z k is larger than 1/r. The functions B 1 X and B 1 S (i.e. B f for f := 1 X and f := 1 S ) satisfy Under the additional assumption (SA), z = 0 is the unique zero of B 1 X (·) in D(0, 1).

V -geometrical ergodicity and bound of the spectral gap
In this section an original proof of the V -geometrical ergodicity of P under the three assumptions (D)-(M)-(SA) is derived from the previous statements. We also provide a new bound of the spectral gap ρ V (P ) of P on B V defined in (5). This bound is related to the real number r ∈ [0, 1) of Theorem 2.1 and to the following real number S only depending on the action of the iterates of P on the small set S in (M): Under Assumptions (D)-(M)-(SA), Proposition 4.1 is used in order to define θ := min |z| : 1 < |z| < 1/r, B 1 X (z) = 0 with the convention θ := 1/r when the previous set is empty. Note that, for every 0 < τ < 1/r, the function B 1 X (·) is analytic on D(0, τ ) from Proposition 4.1 so has a finite number of zeros in D(0, τ ). From this fact and from the definition of θ, it follows that θ > 1. The following lemma is used to derive the inequality S ≤ θ −1 .
Proof. The radius of convergence of σ(z) is larger than 1 since (σ k ) k≥0 is clearly bounded from above by 2 φ V . Next, we deduce from the definitions of T n and π in (7) and (14) that Composing on the left by φ this equality we obtain that ∀n ≥ 1, n k=1 β k (1 X )σ n−k = h n where (h n ) n≥1 is a sequence of complex numbers (depending on φ) such that, for every γ ∈ (r, 1), |h n | = O(γ n ) from Corollary 2.1 and (15). Then Note that h(z) (as B 1 X (z)) has a radius of convergence larger than 1/r since we have, for every γ ∈ (r, 1), |h n | = O(γ n ). Moreover, first z = 0 is the only zero of B 1 X (·) on D(0, θ) from Proposition 4.1 and from the definition of θ, second z = 0 is a simple zero of B 1 X (·) since β 1 (1 X ) = ν(1 X ) > 0. Thus, for every z ∈ D(0, θ), B 1 X (z) = zξ(z) with ξ(z) = +∞ k=0 β k+1 (1 X )z k having a radius of convergence larger than 1/r and having no zero in D(0, θ). It follows from (30) that z → z σ(z) coincides on D(0, 1) with the function h/ξ which is analytic on D(0, θ) since 1/r ≥ θ and ξ does not vanish on D(0, θ). Therefore the power series +∞ k=0 σ k z k+1 has a radius of convergence larger than θ.
We are now in position to state the main result of this section.