We consider continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. This kind of processes are useful in the study of chain molecular diffusions. We give explicit formulas for probability generating functions, and also for means, variances and state probabilities of the random variables of the process. Moreover we study independent random time-changes with the inverse of the stable subordinator, the stable subordinator and the tempered stable subordinator. We also present some asymptotic results in the fashion of large deviations. These results give some generalizations of those presented in [Journal of Statistical Physics 154 (2014), 1352–1364].
Random walks in continuous time are largely employed in several fields of both theoretical and applied interest. In this paper we consider a class of continuous-time Markov chains on integers, called the basic model, which can have transitions to adjacent states only, and with alternating transition rates to their adjacent states; namely we assume to have the same transition rates for the odd states, and the same transition rates for the even states. We also consider some independent random time-changes of the basic model.
Markov chains with alternating rates are useful in the study of chain molecular diffusions. We recall the paper [
In this paper we also consider independent random time-changes of the basic model which provide more flexible versions of the chemical models in the references cited above. More precisely we consider the inverse stable subordinator or, alternatively, the (possibly tempered) stable subordinator. In the first case the particle is subject to a sort of trapping and delaying effect; on the contrary, in the second case, we allow positive jumps in the random time-changed argument, which produces a possible rushing effect.
We start with a more rigorous presentation of the basic model in terms of the generator. In general we consider a continuous-time Markov chain
Transition rate diagram of
We remark that this is a generalization of the model in [
In particular we extend the results in [
Moreover we consider some random time-changes of the basic model
A wide class of random time-changes concerns subordinators, namely nondecreasing Lévy processes (see, for example, [ the inverse of the stable subordinator the (possibly tempered) stable subordinator
In both cases, i.e. for both
We also try to extend the large deviation results for
The applications of the Gärtner Ellis Theorem are based on suitable limits of moment generating functions. So, in view of the applications of this theorem, we study the probability generating functions of the random variables of the processes; in particular the formulas obtained for
There are some references in the literature with applications of the Gärtner Ellis Theorem to time-changed processes. However there are very few cases where the random time-change is given by the inverse of the stable subordinator; see e.g. [
We conclude with the outline of the paper. Section
Some results in this paper concerns the theory of large deviations; so, in this section, we recall some preliminaries (see e.g. [
We also present moderate deviation results. This terminology is used when, for each family of positive numbers
The main large deviation tool used in this paper is the Gärtner Ellis Theorem (see e.g. Theorem 2.3.6 in [
In this section we present the results for the basic model. Some of them will be used for the models with random time-changes in the next sections. We start with some non-asymptotic results, where
In particular the probability generating functions
We also have to consider the function
The non-asymptotic results presented below depend on
The function Λ is the analogue of the function Λ in equation (14) in [ The desired equalities can be checked with some cumbersome computations. Here we only say that it is useful to check the equalities in terms of the function
In this section we present explicit formulas for probability generating functions (see Proposition
In view of this we present some preliminaries. It is known that the state probabilities solve the equations
We start with the probability generating functions.
The main part of the proof consists of the computation of the exponential matrix
The eigenvalues of
We complete the proof noting that, by (
In the next proposition we give mean and variance; in particular we refer to The desired expressions of mean and variance can be obtained with suitable (well-known) formulas in terms of
In this section we present Propositions
We also give some brief comments on the interest of these results (whatever we choose We can simply adapt the proof of Proposition 3.1 in [ We apply the Gärtner Ellis Theorem. More precisely we show that
We remark that
The expressions of mean and variance in Proposition
In this section we consider the process
So, in view of what follows, we recall some preliminaries. We start with the definition of the Mittag-Leffler function (see e.g. [
Now we prove Proposition We recall that
In this section we present Proposition
Finally, in Remark
We want to apply the Gärtner Ellis Theorem and, for all
Firstly, if
Moreover the function
We have some difficulties to get the extension of Proposition
We take for there exists
Thus, by combining these two statements, there exists
We also remark that the statement (
The rate function
In this section we consider the process
So we recall some preliminaries on
Now we prove Proposition
We recall that
We conclude this section with a brief discussion on the condition
In this section we present Propositions
Obviously we can repeat the brief comments on the interest of the results presented just before Propositions
We want to apply the Gärtner Ellis Theorem and, for all
Firstly we have
The function
In view of the next result on moderate deviations we compute We apply the Gärtner Ellis Theorem. More precisely we show that
We remark that
In this paper we study continuous-time Markov chains on integers which allow transitions to adjacent states only, with alternating rates. We present some explicit formulas (means, variances, state probabilities), and we also study the asymptotic behaviour in the fashion of large deviations by applying the Gärtner Ellis Theorem. Moreover we study independent random time-changes of these Markov chains with the inverse of the stable subordinator, the stable subordinator and the tempered stable subordinator. We do not have any large deviation results with the stable subordinator (because we cannot apply the Gärtner Ellis Theorem); on the contrary, when we deal with the tempered stable subordinator, we can provide a complete analysis as we did for the basic model. We also give some large deviation results with the inverse of the stable subordinator but, in this case, we cannot obtain a result on moderate deviations. Some other (possibly dependent) more general random time-changes could be investigated in the future.
In this section we present certain formulas for the state probabilities (
The formulas presented below can be obtained by extracting suitable coefficients of the probability generating functions (see Propositions
In view of what follows we introduce some further notation. Firstly, we write
Finally, in view of the results (Propositions
We conclude with some remarks explaining how to obtain the state probabilities (
Proposition
Proposition
Note that the formulas for the state probabilities (
We thank two anonymous referees for some useful comments which led to an improvement of the presentation.