1. Introduction
We start with the following equation describing a discrete-time motion in 
of a particle with mass 
in the presence of a random potential and a viscosity force proportional to velocity:
Here d-vector 
is the velocity at time 
matrix 
represents an anisotropic damping coefficient, and d-vector 
is a random force applied at time 
The above equation is a discrete-time counterpart of the Langevin SDE 
[1,2]. Applications of the Langevin equation with a random non-Gaussian term 
are addressed, for instance, in [3,4]. Setting 
and 
we obtain:
(1)
The random walk 
associated with this equation is given by
(2)
Similar models of random motion in dimension one, with i.i.d. forces 
were considered in [5-8], see also [9,10] and references therein. See, for instance, [11-14] for interesting examples of applications of Equation (1) with i.d.d. coefficients in various areas.
In this paper we will assume that the coefficients
are induced (in the sense of the following definition) by certain Gibbs’s states.
Definition 1. Coefficients 
are said to be induced by random variables 
each valued in a finite set 
if there exists a sequence of independent random d-vectors 
which is independent of 
and is such that for a fixed
are i.i.d. and 
The randomness of 
is due to two factors:
1) Random environment 
which describes a “state of Nature”; and, given the realization of 
2) The “intrinsic” randomness of systems’ characteristics which is captured by the random variables 
Note that when 
is a finite Markov chain,
is a Hidden Markov Model. See, for instance, [15] for a survey of HMM and their applications. Heavy tailed HMM as random coefficients of multivariate linear time-series models have been considered, for instance, in [16,17]. In the context of financial time series, 
can be interpreted as an exogenous factor determined by the current state of the underlying economy. The environment changes due to seasonal effects, response to the news, dynamics of the market, etc. When 
is a function of the state of a Markov chain, stochastic difference Equation (1) is a formal analogue of the Langevin equation with regime switches, which was studied in [18]. The notion of regime shifts or regime switches traces back to [19,20], where it was proposed in order to explain the cyclical feature of certain macroeconomic variables.
In this paper we consider 
that belong to the following class of random processes:
Definition 2 ([21]). A C-chain is a stationary random process 
taking values in a finite set (alphabet)
such that the following holds:
i) For any 
ii) For any 
and any sequence 
the following limit exists:
where the right-hand side is a regular version of the conditional probabilities.
iii) Let
Then, 
C-chains form an important subclass of chains with complete connections/chains of in-finite order [22-24]. They can be described as exponentially mixing full shifts, and alternatively defined as an essentially unique random process with a given transition function (g-measure) 
[25]. Stationary distributions of these processes are Gibbs states in the sense of Bowen
[21,26]. For any C-chain 
there exists a Markovian representation [21,25], that is a stationary irreducible Markov chain 
in a countable state space and a function 
such that 
where 
means equivalence of distributions. Chains of infinite order are well-suited for modeling of long range-dependence with fading memory, and in this sense constitute a natural generalization of finite-state Markov chains [24,27-30].
We will further assume that the vectors 
are multivariate regularly varying. Recall that, for 
a function 
is said to be regularly varying of index 
if 
for some function
such that 
for any positive real 
(i.e., 
is a slowly varying function). Let
Definition 3 ([31]). A random vector 
is regularly varying with index 
if there exist a function 
regularly varying with index 
and a Radon measure 
in the space 
such that
as 
where 
denotes the vague convergence and 
We denote by 
the set of all d-vectors regularly varying with index 
associated with function 
The corresponding limiting measure 
is called the measure of regular variation associated with 
We next summarize our assumptions on the coefficients 
and 
Let 
and
for, respectively, a vector 
and a 
matrix
Assumption 1. Let 
be a stationary C-chain defined on a finite state space 
and suppose that 
is induced by 
Assume in addition that:
A1) 
where 
for 
A2) The spectral radius 
is strictly between zero and one.
A3) There exist a constant 
and a regularly varying function 
with index 
such that for all 
with associated measure of regular variation 
2. Statement of Results
For any (random) initial vector 
the series 
converges in distribution, as 
to
which is the unique initial value making 
into a stationary sequence [32]. The following result, whose proof is omitted, is a “Gibssian” version of a “Markovian” [16, Theorem 1]. The claim can be established following the line of argument in [16] nearly verbatim, exploiting the Markov representation of C-chains obtained in [21].
Theorem 1. Let Assumption 1 hold. Then 
with associated measure of regular variation
where 
stands for 
and 
In a slightly more general setting, the existence of the limiting velocity suggests the following law of large numbers, whose short proof is included in Section 3.1.
Theorem 2. Let Assumption 1 hold with A3) being replaced by the condition 
Then1
, a.s.
Let 
denote independent copies of 
and let be 
a sequence of vectors such that the sequence of processes
converges in law as 
in the Skorokhod space
to a Lévy process 
where 
are introduced in A3) with stationary independent increments, 
and 
being distributed according to a stable law of index 
whose domain of attraction includes 
For an explicit form of the centering sequence 
and the characteristic function of 
see, for instance, [33] or [34]. Remark that one can set 
if 
and 
if
For each 
define a process 
in
by setting
(3)
Theorem 3. Let Assumption 1 hold with 
Then the sequence of processes 
converges weakly in 
as 
to 
It follows from Definition 3 (see, for instance, [31])
that if 
then the following limit exists for any vector 
(4)
Theorem 4. Assume that the conditions of Theorem 3 hold. If 
assume in addition that the law of 
is symmetric for any 
Let 
be defined by Equation (4) with 
Then, for any 
such that 
we have
a.s. (5)
In particular,
a.s.
If either Assumption 1 holds with 
or
is assumed instead of A3), then, in view of Equation (6), a Gaussian counterpart of Theorem 3 can be obtained as a direct consequence of general CLTs for uniformly mixing sequences (see, for instance, [35, Theorem 20.1] and [36, Corollary 2]) applied to the sequence 
If 
then a law of iterated logarithm in the usual form follows from Equation (5) and, for instance, [37, Theorem 5] applied to the sequence 
We remark that in the case of i.i.d. additive component 
similar to our results are obtained in [7] for a more general than Equation (1) mapping 
3. Proofs
3.1. Proof of Theorem 2
It follows from the definition of the random walk 
and Equation (1) that
(6)
Note that 
implies
It follows then from the Borel-Cantelli lemma that
a.s.
Furthermore, we have
Thus the law of large numbers for 
follows from the ergodic theorem applied to the sequence 
□
3.2. Proof of Theorem 3
Only the second term in the right-most side of Equation (5) contributes to the asymptotic behavior of 
The proof rests on the application of Corollary 5.9 in [34] to the partial sums 
In view of condition iii) in Definition 2 and the decomposition shown in Equation (6), we only need to verify that the following “local dependence” condition (which is condition (5.13) in [34]) holds for the sequence 
The above convergence to zero follows from the mixing condition iii) in Definition 2 and the regular variation, as t goes to infinity, of the marginal distribution tail
□
3.3. Proof of Theorem 4
For 
let 
be the number of occurrences of 
in the set 
That is,
Define recursively 
and
(with the usual convention that the greatest lower bound over an empty set is equal to infinity). For
let
where
Denote
Further, for each 
let 
if 
whereas if 
let
Then 
and hence
It follows from the decomposition given by Equation (6) along with the Borel-Cantelli lemma that for any 
a.s.
Let 
Then
It follows, for instance, from Theorem 5 in [37] that if 
then for any 
the following limit exists and the identity holds with probability one:
Therefore (since 
is regularly varying with index
), in order to complete the proof Theorem 4 it suffices to show that for any 
that satisfies the condition 
of the theorem, we have
a.s.
We first observe that by the law of iterated logarithm for heavy-tailed i.i.d. sequences (see Theorems 1.6.6 and 3.9.1 in [33]),
, a.s.
for any 
and 
Since by the ergodic theorem,
a.s.this yields
, a.s.and hence
, a.s.
On the other hand, if 
Theorem 3.9.1 in [33] implies that for any 
and any 
such that 
we have
a.s.
To conclude the proof of the theorem it thus remains to show that for any 
any
and all 
(7)
where, for 
the events 
are defined as follows:
.
For 
let 
and define
.
Then
The Ruelle-Perron-Frobenius theorem (see [26]) implies that the sequence 
satisfies the large deviation principle (by the Gärtner-Ellis theorem), and hence
for some constants 
and 
Furthermore, for any 
and 
there exists a constant 
such that (see [33, p. 177]), 
Therefore, since 
we can choose 
such that 
with suitable 
and 
A standard argument using the Borel-Cantelli lemma imply then the identity in Equation (7). □