Limit theorems for affine Markov walks conditioned to stay positive
Abstract.
Consider the real Markov walk with increments defined by a stochastic recursion starting at . For a starting point denote by the exit time of the process from the positive part of the real line. We investigate the asymptotic behaviour of the probability of the event and of the conditional law of given as .
Key words and phrases:
Exit time, stochastic recursion, Markov chains, harmonic function2000 Mathematics Subject Classification:
Primary 60J05, 60J50, 60G50. Secondary 60J70, 60G42Grama, I.] Université de BretagneSud, LMBA UMR CNRS 6205, Vannes, France
Lauvergnat, R.]Université de BretagneSud, LMBA UMR CNRS 6205, Vannes, France
Le Page, É.]Université de BretagneSud, LMBA UMR CNRS 6205, Vannes, France
1. Introduction
Assume that the Markov chain is defined by the stochastic recursion
(1.1) 
where is a sequence of i.i.d. real random pairs satisfying for some . Consider the Markov walk . Under a set of conditions ensuring the existence of the spectral gap of the transition operator of the Markov chain , it was established in Guivarc’h and Le Page [17] that there exist constants and such that, for any ,
(1.2) 
where is the standard normal distribution function and is the probability measure generated by starting at . There are easy expressions of and in terms of law of the pair : in particular .
For a starting point , define the first time when the affine Markov walk becomes nonpositive by setting
In this paper we complete upon the results in [17] by determining the asymptotic of the probability and proving a conditional version of the limit theorem (1.2) for the sum , given the event in the case when The main challenge in obtaining these asymptotics is to prove the existence of a positive harmonic function pertaining to the associated Markov chain . A positive harmonic function, say , is defined as a positive solution of the equation , where is the restriction on of the Markov transition kernel Q of the chain .
From the more general results of the paper it follows that, under the same hypotheses that ensure the CLT (see Condition 1 in Section 2), if the pair is such that and , for some , then
and
where is the Rayleigh distribution function. In particular, the above mentioned results hold true if and are independent and is such that and . Less restrictive assumptions on the pair are formulated in our Section 2.
The above mentioned results are in line with those already known in the literature for random walks with independent increments conditioned to stay in limited areas. We refer the reader to Iglehart [18], Bolthausen [2], Doney [11], Bertoin and Doney [1], Borovkov [4, 3], Caravenna [5], Eichelsbacher and Köning [12], Garbit [13], Denisov, Vatutin and Wachtel [7], Denisov and Wachtel [8, 10]. More general walks with increments forming a Markov chain have been considered by Presman [20, 21], Varapoulos [22, 23], Dembo [6], Denisov and Wachtel [9] or Grama, Le Page and Peigné [15]. In [20, 21] the case of sums of lattice random variables defined on finite regular Markov chains has been considered. Varapoulos [22, 23] studied Markov chains with bounded increments and obtained lower and upper bounds for the probabilities of the exit time from cones. Some studies take advantage of additional properties: for instance in [9] the Markov walk has a special integrated structure; in [15] the moments of are bounded by some constants not depending on the initial condition. However, to the best of our knowledge, the asymptotic behaviour of the probability in the case of the stochastic recursion (1.1) has not yet been considered in the literature.
Note that the WienerHopf factorization, which usually is employed in the case of independent random variables, cannot be applied in a straightforward manner for Markov chains. Instead, to study the case of the stochastic recursion, we rely upon the developments in [9], [10] and [15]. The main idea of the paper is given below. The existence of the positive harmonic function is linked to the construction of a martingale approximation for the Markov walk . While the harmonicity is inherently related to the martingale properties, the difficulty is to show that the approximating martingale is integrable at the exit time of the Markov walk . In contrast to [10] and [15], our proof of the existence of employs different techniques according to positivity or not of the values of . The constructed harmonic function allows to deduce the properties of the exit time and the conditional distribution of the Markov walk from those of the Brownian motion using a strong approximation result for Markov chains from Grama, Le Page and Peigné [16]. The dependence of the constants on the initial state of the Markov chain established there plays the essential role in our proof.
The technical steps of the proofs are as follows. We first deal with the case when the starting point of the Markov walk is large: for some . When is arbitrary, the law of iterated logarithm ensures that the sequence will cross the level with high probability. Then, by the Markov property, we are able to reduce the problem to a Markov walk with a large starting point , where is the first time when the sequence exceeds the level . The major difficulty, compared to [10] and [15], is that, for the affine model under consideration, the sequence is not bounded in . To overcome this we need a control of the moments of in function of the initial state and the lag
We end this section by agreeing upon some basic notations. As from now and for the rest of this paper the symbols denote positive constants depending only on their indices. All these constants are likely to change their values every redoccurrence. The indicator of an event is denoted by For any bounded measurable function on , random variable in and event , the integral means the expectation .
2. Notations and results
Assume that on the probability space we are given a sequence of independent real random pairs , , of the same law as the generic random pair . Denote by the expectation pertaining to . Consider the Markov chain defined by the affine transformations
where is a starting point. The partial sum process defined by for all and will be called in the sequel affine Markov walk. Note that itself is not a Markov chain, but the pair forms a Markov chain.
For any , denote by the transition probability of . Introduce the transition operator
which is defined for any real bounded measurable function on . Denote by and the probability and the corresponding expectation generated by the finite dimensional distributions of starting at . Clearly, for any and , we have .
We make use of the following condition which ensures that the affine Markov walk satisfies the central limit theorem (1.2) (c.f. [17]):
Condition 1.
The pair is such that:

There exists a constant such that and

The random variable is nonzero with positive probability, , and centred, .
Note that Condition 1 is weaker than the conditions required in [17] in the special case . Nevertheless, using the same techniques as in [17] it can be shown that, under Condition 1, the Markov chain has a unique invariant measure and its partial sum satisfies the central limit theorem (1.2) with
(2.1) 
and
(2.2) 
Moreover, it is easy to see that under Condition 1 the Markov chain has no fixed point: , for any . Below we make use of a slightly refined result which gives the rate of convergence in the central limit theorem for with an explicit dependence of the constants on the initial value stated in Section 9.3.
For any consider the affine Markov walk starting at and define its exit time
Corollary 9.7 implies the finiteness of the stopping time : under Condition 1, it holds for any and .
The asymptotic behaviour of the probability is determined by the harmonic function which we proceed to introduce. For any , denote by the transition probability of the Markov chain The restriction of the measure on is defined by
for any measurable set on and for any . Let be a measurable set in containing . For any measurable set harmonic function on is any function which satisfies A positive
To ensure the existence of a positive harmonic function we need additional assumptions:
Condition 2.
For all and ,
Condition 3.
For any and , there exists such that for any constant , there exists such that,
where
Obviously Condition 2 is equivalent to for any and , which, in turn is equivalent to the fact that there exists such that , for any and . Therefore Condition 3 implies Condition 2. As a byproduct, under either Condition 2 or Condition 3, the event is not empty.
The existence of a harmonic function is guaranteed by the following theorem. For any consider the martingale defined by
(2.3) 
with the natural filtration (we refer to Section 3 for details).
Theorem 2.1.
Assume either Conditions 1, 2 and , or Conditions 1 and 3.

For any and , the random variable is integrable,
and the function
is well defined on .

The function is positive and harmonic on : for any and ,

Moreover, the function has the following properties:

For any , the function is nondecreasing.

For any , , and ,

For any , it holds

Using the harmonic function from the previous theorem, we obtain the asymptotic of the tail probability of the exit time .
Theorem 2.2.
Moreover, we prove that the Markov walk conditioned to stay positive satisfies the following limit theorem.
Theorem 2.4.
Theorem 2.5.
Below we discuss two more restrictive assumptions which, however, are easier to verify than Conditions 2 and 3, respectively.
Condition 2bis.
The law of the pair is such that for all ,
Condition 3bis.
There exists such that,
It is straightforward that Condition 2bis implies Condition 2. This follows from the inequality
with . The fact that Condition 3bis implies Condition 3 is proved in the Appendix 9.1.
Under Condition 1, it is easy to see that Condition 3bis is satisfied, for example, when random variables and are independent and and .
Note that, while Condition 3 implies Condition 2, there is no link between Conditions 2bis and 3bis. Indeed, if and are independent, is nonnegative and the support of contains , then Condition 2bis holds true whereas Condition 3bis does not. At the opposite, if and are independent bounded and support of equal to then Condition 3bis holds true whereas Condition 2bis does not.
The outline of the paper is as follows. The martingale approximation of the Markov walk and some of its properties are given in Section 3. In Section 4 we prove that the expectation of the killed Markov walk is bounded uniformly in . This allows us to prove the existence of the harmonic function and establish some of its properties in Section 5. With the help of the harmonic function and of a strong approximation result for Markov chains we prove Theorems 2.2, 2.4 and 2.5, in Sections 6, 7 and 8 respectively. Section 9 is an appendix where we collect some results used in the proofs.
3. Martingale approximation
In this section we approximate the Markov walk by a martingale following Gordin [14]. We precede this construction by a lemma which shows that there is an exponential decay of the dependence of on the initial state as grows to infinity. This simple fact will be used repeatedly in the sequel.
Lemma 3.1.
For all , , and ,
Proof.
Since , for , with the convention , we have by the Minkowski inequality and the independence of ,
The conclusion of the lemma is thus a direct consequence of Condition 1. ∎
Let , be the identity function on . The Poisson equation has a unique solution , given by,
Using the function , the process defined in (2.3) can be recast as
Consider the natural filtration with the trivial algebra and the algebra generated by The fact that is indeed a martingale, for any , is easily verified by the Markov property: for
All over the paper we use the abbreviation
(3.1) 
With this notation, for any and , the Markov walk has the following martingale representation:
(3.2) 
Define the sequence , by
(3.3) 
with the convention for . The sequence corresponds to the stochastic recursion starting at . In the same line, we define and , for all . It is easy to see that the process is a zero mean martingale which is related to the martingale by the identity
(3.4) 
where
The following two lemmas will be used to control .
Lemma 3.2.

The sequence is a centred martingale.

For all and ,
Proof.
The first claim follows from the fact that is a difference of two martingales. Using the Minkowski inequality for , the independence of and Condition 1 we obtain the second claim. ∎
Let us introduce the martingale differences:
Lemma 3.3.
For all and ,
Proof.
For the increments we simply use Lemma 3.1 with . For the martingale , the upper bound is obtained by Burkholder inequality: for all and all ,
By the Hölder inequality with the exponents and , we obtain
This proves the claim when . When the assertion follows obviously using Jensen inequality. ∎
Lemma 3.4.
For all and ,
4. Integrability of the killed martingale
The goal of this section is to prepare the background to prove the integrability of the random variable which is crucial for showing the existence of the harmonic function in Section 5. We use different approaches depending on the sign on : when , in Section 4.2 we prove that the expectation of the martingale killed at is uniformly bounded in , while, when , in Section 4.3 we prove that the expectation of the same martingale killed at is uniformly bounded in , where is the exit time of the martingale .
4.1. Preliminary results
We first state a result concerning the first time when the process (respectively ) crosses the level . Introduce the following stopping times: for any , , and ,
and
Lemma 4.1.
Let . There exists such that for any , , , and ,
and
Proof.
With , where is defined in Corollary 9.6 and a constant to be chosen below, let , and for any , and , with ,
Note that by the martingale representation (3.2), we have for any , . Then, choosing large enough to have ,
Moreover, we have also,
Since is a Markov chain,
(4.1) 
We use the decomposition (3.4) to write that, with ,
Using (3.2) with , we have . By the Markov inequality,
Since does not depend on , using Lemma 3.1 and the claim 2 of Lemma 3.2, we obtain
Inserting this bound in (4.1), it follows that
Set . Denote by the closed ball centred in of radius . The rate of convergence in the central limit theorem from Corollary 9.6 (applied with ) implies that,
Moreover,
Let . With large enough in the definition of , we have and thus ,
Iterating, we get
Using the fact that
∎
4.2. Integrability of the killed martingale: the case
The difficulty in proving that the expectation