GEOMETRIC TRANSIENCE FOR NON-LINEAR AUTOREGRESSIVE MODELS

SONG Yan-hong

(School of Statistics and Mathematics，Zhongnan University of Economics and Law，Wuhan 430073，China）

GEOMETRIC TRANSIENCE FOR NON-LINEAR AUTOREGRESSIVE MODELS

SONG Yan-hong

(School of Statistics and Mathematics，Zhongnan University of Economics and Law，Wuhan 430073，China）

In the paper，we study the stochastic stability for non-linear autoregressive models.By establishing an appropriate Foster-Lyapunov criterion，a sufficient condition for geometric transience is presented.

geometric transience；non-linear autoregressive model；Foster-Lyapunov criterion

2010 MR Subject Classification：60J05；60J20；37B25

Document code：AArticle ID：0255-7797（2016）05-0987-06

1 Introduction

Consider a non-linear autoregressive Markov chainondefined by

First，let us recall some notations and definitions，see［3，5，6］for details.Denote by）the Borel σ-field on R，and write=.The n-steptransition kernel of the chain Φnis defined as

The chain Φnis Lebesgue-irreducible，if for every

Obviously，the subset of a petite set is still petite.By［7，Lemma 2.1］or［8，Theorem 1］，we know that the non-linear autoregressive model is Lebesgue-irreducible，and every compact set inis petite.

be the first return and first hitting times，respectively，on A.It is obvious that τA=σAif Φ0∈Ac.Denote by L（x，A）=the probability of the chain Φnever returning to A.

The chain Φnis called geometrically transient，if it is ψ-irreducible for some non-trivial measure ψ，and R can be covered ψ-a.e.by a countable number of uniformly geometrically transient sets.That is，there exist sets D and Ai，i=1，2，···such that=D∪where ψ（D）=0 and each Aiis a uniformly geometrically transient set of the chain Φn.

To state the main result of this paper，we need the following assumptions：

Theorem 1.1Assume（A1）and（A2）.Then the non-linear autoregressive model Φnis geometrically transient.

Remark 1.2It is easy to see that（A2）is equivalent to the condition in［9，Theorem 3.1］，where transience for the the non-linear autoregressive model Φnwas confirmed.Here，we get a stronger result（i.e.geometric transience）in Theorem 1.1.

2 Proof of Theorem 1.1

This section is devoted to proving Theorem 1.1 by using the Foster-Lyapunov（or drift）condition for geometric transience.

It is well known that Foster-Lyapunov conditions were widely used to study the stochastic stability for Markov chains.For examples，Down，Meyn and Tweedie［10-13］studied the drift conditions for recurrence，ergodicity，geometric ergodicity and uniform ergodicity.The drift conditions for sub-geometric ergodicity were discussed in［1，4，14-17］and so on.In ［18，19］，the drift conditions for transience were obtained.

Recently，we investigated the drift condition for geometric transience in［6］.One of the main results shows that the chain Φnis geometrically transient，if there exist some set，constants λ，b∈（0，1），and a function W≥1A（with W（x0）＜∞for somesatisfying the drift condition

As far as we know，however，this drift condition can not be applied directly for the nonlinear autoregressive model considered in this paper.Alternatively，we will establish a more practical drift condition for geometric transience.First，we need the following two lemmas，which are taken from［6］.

Lemma 2.1The chain Φnis geometrically transient if and only if there exist someand a constant κ＞1 such that

Proposition 2.3The chain Φnis geometrically transient，if there exist a petite set，constants λ∈（0，1），b∈（0，∞），and a non-negative measurable function W bounded on A satisfying

and

ProofSince W is non-negative and），we have.Set

Hence by the comparison theorem of the minimal non-negative solution（see［20，Theorem 2.6］），we know from（2.3）and（2.4）that

By（2.5）and noting that D?Ac，we have for all x∈R，

Thus there exists some set C?A withsuch that

According to Lemma 2.1，in the following，it is enough to prove that for some κ＞1，

Combining Lemma 2.2（1）with（2.5）and（2.1），we get for all x∈A，

Since W is bounded on A，

Noting that A is petite and C?A，according to（2.7）and the proof of［3，Theorem 15.2.1］，we obtain that for all 1＜κ≤λ-1/2＜∞.This together with（2.6） yields the desired assertion.

Now，we are ready to prove Theorem 1.1.

Proof of Theorem 1.1By（A2），there exist constants θ＞0 and c＞0 satisfying

Choose

That is，

Noting that W is bounded，it is obvious that for some b∈（0，∞），

Combining this with（2.9），the drift condition（2.1）holds.Thus the non-negative autoregressive model Φnis geometrically transient by Proposition 2.3.

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非線性自回歸模型的幾何非常返性

宋延紅
（中南財經(jīng)政法大學(xué)統(tǒng)計與數(shù)學(xué)學(xué)院，湖北武漢430073）

本文研究了非線性自回歸模型的隨機(jī)穩(wěn)定性.通過建立恰當(dāng)?shù)腇oster-Lyapunov條件，得到了非線性自回歸模型幾何非常返的充分條件.

幾何非常返；非線性自回歸模型；Foster-Lyapunov條件

MR（2010）主題分類號：60J05；60J20；37B25O211.62；O211.9

date：2016-03-11Accepted date：2016-06-01

Supported by National Natural Science Foundation of China（11426219；11501576）.

Biography：Song Yanhong（1983-），female，born at Yantai，Shandong，lecturer，major in probability.