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

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

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.