Noncentral moderate deviations for fractional Skellam processes

The term \emph{moderate deviations} is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. We talk about \emph{noncentral moderate deviations} when the weak convergence is towards a non-Gaussian distribution. In this paper we present noncentral moderate deviation results for two fractional Skellam processes in the literature (see Kerss, Leonenko and Sikorskii, 2014). We also establish that, for the fractional Skellam process of type 2 (for which we can refer the recent results for compound fractional Poisson processes in Beghin and Macci (2022)), the convergences to zero are usually faster because we can prove suitable inequalities between rate functions.


Introduction
A large deviation principle provides some asymptotic bounds for a family of probability measures on the same topological space X ; moreover one often refers to a family of X -valued random variables, {C t }, whose laws are those probability measures.These asymptotic bounds are expressed in terms of a speed function v t (that tends to infinity) and a lower semicontinuous rate function I : X → [0, ∞].The concept of large deviation principle is a basic definition in the theory of large deviations; this theory allows us to compute the probabilities of rare events on an exponential scale (see [10] as a reference of this topic).
The term moderate deviations is often used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between two asymptotic regimes: 1. the convergence of C t in probability to zero, which is governed by a large deviation principle with speed v t ; 2. the weak convergence of √ v t C t to a centered Normal distribution.
The speed functions and the random variables involved in these large deviation principles depend on some scalings in a suitable class; moreover, the large deviation principles in this class are governed by the same quadratic rate function that uniquely vanishes at zero.Typically the scalings consist of families of positive numbers {a t : t > 0} such that a t → 0 and v t a t → ∞, and one can show that { √ v t a t C t } satisfies the large deviation principle with speed 1 at ; note that 1 at has a lower intensity than the speed v t , and this explains the use of the term moderate.We also recall that we recover the two asymptotic regimes stated above for a t = 1 vt (in this case v t a t → ∞ fails) and for a t = 1 (in this case a t → 0 fails).
The term noncentral moderate deviations has been recently used in the literature when we have a class of large deviation principles that, in some sense, fills the gap between a convergence to a constant (typically zero) and the weak convergence towards a non-Gaussian distribution.Some examples of noncentral moderate deviations can be found in [12], where the weak convergences are towards Gumbel, exponential, and Laplace distributions.In that reference, the interested reader can find some other previous references in the literature with some other examples.
The aim of this paper is to present some examples of noncentral moderate deviations based on fractional Skellam processes.In these examples we always have v t = t, and therefore the scalings are families of positive numbers {a t : t > 0} such that a t → 0 and ta t → ∞. ( Skellam processes are given by the difference of two independent Poisson processes.In this paper we consider two fractional Skellam processes studied in [20]; some more recent generalized versions of these processes can be found in [14] and [17].The fractional Skellam processes in [20] are closely related to the definition of the fractional Poisson process in the literature.We recall that a fractional Poisson process is obtained as an independent random time-change of a Poisson process with an inverse of stable subordinator (see e.g.[5], [6] and [22]); here we are referring to the time fractional Poisson process and, for the definitions of space and space-time fractional Poisson process, the interested reader can refer to [23] (see also [19] as a very recent paper on time-changed space-time fractional Poisson processes).Then the fractional Skellam processes studied in [20] are obtained in a quite natural way as follows.
• Fractional Skellam process of type 1.A difference between two independent fractional Poisson processes (so we have two independent random time-changes for each one of the involved fractional Poisson processes); • Fractional Skellam process of type 2.An independent random time-change of a Skellam process with an inverse of stable subordinator.
It is easy to check (see Remark 2.3) that the fractional Skellam process of type 2 is a particular compound fractional Poisson process (and this is not surprising because a Skellam process is a particular compound Poisson process; see Remark 2.1).Therefore, the moderate deviation results for the fractional Skellam process of type 2 can be obtained from the ones in [3].Here, since we deal with random time-changes of Skellam processes, for completeness we recall the references [7], [8] and [18].
Here for completeness we present a brief review of the references with results on large/moderate deviations for fractional Poisson processes or similar models: [1] and [2] with results for the (possibly multivariate) alternative fractional Poisson process, [4] with results for random time-changed continuous-time Markov chains on integers with alternating rates, [21] with results for a nonstandard model based on the Prabhakar function in [24], and for a state dependent model in [11].
We conclude with the outline of the paper.We start with some preliminaries in Section 2. The results for the fractional Skellam processes of type 1 and 2 are presented in Sections 3 and 4, respectively.In Section 5 we compare some rate functions and present some plots.Finally, in Section 6, we present some concluding remarks.

Preliminaries
In this section, we recall some preliminaries on large deviations and on fractional Skellam processes.

On large deviations
Here we present definitions and results for families of real random variables {Z t : t > 0} defined on the same probability space (Ω, F, P ); moreover, in view of what follows, we consider the case t → ∞.We start with the definition of large deviation principle (see e.g.[10]  Then, if Λ is essentially smooth and lower semi-continuous, then {Z t : t > 0} satisfies the LDP with speed v t and good rate function Λ * . We also recall (see e.g.Definition 2.3.5 in [10]) that Λ is essentially smooth if the interior of D(Λ) is non-empty, the function Λ is differentiable throughout the interior of D(Λ), and Λ is steep, i.e. |Λ ′ (θ n )| → ∞ whenever {θ n : n ≥ 1} is a sequence of points in the interior of D(Λ) which converge to a boundary point of D(Λ).A particular simple case (which always occurs in the applications of the Gärtner Ellis Theorem in this paper) is when D(Λ) = R and Λ is a differentiable function; indeed, in such a case, the function Λ is essentially smooth (the steepness condition holds vacuously) and lower semi-continuous.

On fractional Skellam processes
We start with the definition of the (non-fractional) Skellam process.Let {N λ 1 (t) : t ≥ 0} and {N λ 2 (t) : t ≥ 0} be two independent Poisson processes with intensities λ 1 > 0 and λ 2 > 0, respectively.In particular we consider the notation λ = (λ 1 , λ 2 ).Then the process {S λ (t) : t ≥ 0} defined by is called Skellam process.Moreover, for each fixed t ≥ 0, we have Remark 2.1.It is easy to check that {S λ (t) : t ≥ 0} can be seen as a compound Poisson process and In view of what follows we recall some other preliminaries.We start with the definition of the Mittag-Leffler function (see e.g.[13], eq.(3.1.1)) Actually throughout this paper we have ν ∈ (0, 1) and x ∈ R; moreover it is known (see Proposition 3.6 in [13] for the case α ∈ (0, 2); indeed α in that reference coincides with ν in this paper) that we have (this is the correct version of eq. ( 3) in [3]; indeed we need the condition presented here for x → −∞, instead of E ν (x) → 0).Now we recall some moment generating functions which can be expressed in terms of the Mittag-Leffler function.If we consider the inverse of the stable subordinator {L ν (t) : t ≥ 0}, then we have This formula appears in several references with θ ≤ 0 only; however, this restriction is not needed because we can refer to the analytic continuation of the Laplace transform with complex argument.The fractional Poisson process {N ν,λ (t) : t ≥ 0} is defined by where Now we are ready to provide the definitions of two fractional Skellam processes and their moment generating functions (see [20], Definitions 3.1-3.2and Theorems 3.1-3.2).In particular we consider the notation ν = (ν 1 , ν 2 ) for ν 1 , ν 2 ∈ (0, 1).
Fractional Skellam process of type 2. It is the process {Z ν,λ (t) : t ≥ 0} defined by where the Skellam process {S λ (t) : t ≥ 0} and the inverse of the stable subordinator {L ν (t) : t ≥ 0} are independent.Then we have Remark 2.3.We have recalled that the fractional Poisson process can be seen as a time changed (non-fractional) Poisson process with an independent inverse of the stable subordinator.Then, by taking into account Remark 2.1, we can say that {Z ν,λ (t) : t ≥ 0} is distributed as the compound fractional Poisson process Remark 2.4.Assume that ν 1 = ν 2 = ν for some ν ∈ (0, 1) (and recall the slight change of notation explained in Remark 2.2 for the fractional Skellam process of type 1).Then, if λ 1 = λ 2 , the random variables Y ν,λ (t) and Z ν,λ (t) are symmetric (around zero); namely Y ν,λ (t) and Z ν,λ (t) are distributed as −Y ν,λ (t) and −Z ν,λ (t), respectively.Then we have some consequences highlighted in Remarks 3.3 and 4.3.

Noncentral moderate deviations for the type 1 process
We start with the first result for which we could have ν,λ be the function defined by Then Y ν,λ (t) t : t > 0 satisfies the LDP with speed v t = t and good rate function I LD defined by Proof.We prove this proposition by applying the Gärtner Ellis Theorem.More precisely we have to show that lim where Ψ ν,λ is the function in (4).The case θ = 0 is immediate.Moreover, we remark that Then, by taking into account the asymptotic behaviour of the Mittag-Leffler function in (2), we have and Thus the limit in ( 6) is checked.
In conclusion the desired LDP holds noting that the function Ψ ν,λ in (4) is finite (for all θ ∈ R) and differentiable.
Proof.We have to check that (for all θ ∈ R) (here we take into account that L • ν (1) and L •• ν (1) are i.i.d., and the expression of the moment generating function in eq. ( 3)).This can be readily done noting that, for two suitable remainders o 1 t 2ν such that t 2ν o 1 t 2ν → 0, we have ; then we get the desired limit letting t go to infinity (for each fixed θ ∈ R).
Then, for every family of positive numbers {a t : t > 0} such that (1) holds, the family of random variables : t > 0 satisfies the LDP with speed 1/a t and good rate function I (1) MD defined by Proof.We prove this proposition by applying the Gärtner Ellis Theorem.More precisely we have to show that where Ψ(1) ν,λ is the function defined by MD in the statement of the proposition (for x = 0 the supremum is attained at θ = 0, for x > 0 the supremum is attained at θ ).So we conclude the proof by checking the limit in (8).The case θ = 0 is immediate.Moreover we remark that, for two suitable remainders o .
Then, by taking into account the asymptotic behaviour of the Mittag-Leffler function in (2), we have and Thus the limit in ( 8) is checked.
Remark 3.2.The set {x ∈ R : I LD is not available, because ν,λ is a symmetric function); the weak limit in Proposition 3.2 is a symmetric random variable; the rate function MD in Proposition 3.3 is a symmetric function.

Noncentral moderate deviations for the type 2 process
The results in this section can be derived directly from the results in [3]; indeed, by taking into account Remark 2.3, the fractional Skellam process of type 2 is a particular compound fractional Poisson process.So we only give the statements of propositions without proofs.
Then Z ν,λ (t) t : t > 0 satisfies the LDP with speed v t = t and good rate function I LD defined by ν,λ (θ)}.
where, after some computations, one can check that The next two propositions can be derived from Propositions 3.2-3.3 in [3].By Remark 2.3 we X k , where coincide with µ and σ 2 in [3].Moreover we set which coincides with α(ν) in [3].Also note that, if λ 1 = λ 2 = λ for some λ > 0, we take into account that Proposition 4.2.Let α 2 (ν) be defined in (10).Then: t : t > 0} converges weakly to 2λL ν (1)W , where W is a standard Normal distributed random variable, and independent to L ν (1); Proposition 4.3.Let α 2 (ν) be defined in (10).Then, for every family of positive numbers {a t : t > 0} such that (1) holds, the family of random variables : t > 0 satisfies the LDP with speed 1/a t and good rate function I MD,λ defined by: MD,λ (x) := Remark 4.2.The sets {x ∈ R : I MD,λ (x) < ∞} (see Proposition 4.3) coincide with the supports of the weak limits in Proposition 4.2: we mean Remark 4.3.Assume that λ 1 = λ 2 .Then: the rate function LD in Proposition 4.1 is a symmetric function (we can say this, even if an explicit expression of ν,λ is a symmetric function); the weak limit in Proposition 4.2 is a symmetric random variable; the rate function MD in Proposition 4.3 is a symmetric function.

Comparisons between rate functions
In this section we compare the rate functions for the two types of fractional Skellam process, and for different values of ν.Moreover, we present some plots.

Results and remarks
All the LDPs presented in the previous sections are governed by rate functions that uniquely vanish at x = 0. So, as we explain below, it is interesting to compare the rate functions, at least around x = 0. Remark 5.1.Throughout this section we always assume that ν 1 = ν 2 = ν for some ν ∈ (0, 1).So we simply write Ψ Proof.We start noting that, for the function Ψ The first statement in (11) is immediate.For the second statement we have two cases (for completeness we remark that λ 1 (e θ − 1) + λ 2 (e −θ − 1) ≥ 0 for all θ ∈ R if λ 1 = λ 2 ).
• Assume that λ 1 < λ 2 .Then for x > 0 we have I MD (x) < ∞ = I MD,λ (x).For x < 0 we have , which is trivially equivalent to I MD (x) < I MD,λ (x).Finally, if λ 1 = λ 2 = λ for some λ > 0, the statement to prove trivially holds noting that, for two constants c ν,λ > 0, we have Proposition 5.1 tells us that if we compare the rate functions in Propositions 3.1 and 4.1, the rate function of the fractional Skellam process of type 2 is larger than that of the fractional Skellam process of type 1. Proposition 5.2 tells us the same for the rate functions in Propositions 3.3 and 4.3 but, when λ 1 = λ 2 , the rate function of the fractional Skellam process of type 2 is larger only around x = 0.
These inequalities between rate functions allow us to say that the convergence of random variables for the fractional Skellam process of type 2 is faster than the corresponding convergence for the fractional Skellam process of type 1.We explain this by considering the LDPs in Proposition 3.1, with ν 1 = ν 2 = ν for some ν ∈ (0, 1), and in Proposition 4.1.Indeed, for every δ > 0 we have lim LD (δ), where J LD (δ) := min{I LD (δ), I LD (−δ)} 2.1) which can be applied because the involved moment generating functions are available in a closed form given in terms of the Mittag-Leffler function.
For the classical (non-fractional) Skellam process we can obtain a classical moderate deviation result that fills the gap between the two following asymptotic regimes: In general the applications of the Gärtner Ellis Theorem for random time-changed processes are quite standard.However, in order to obtain noncentral moderate deviation results as the ones  in this paper, it is important to have a random time-change with a slowing effect as happens in this paper with the inverse of the stable subordinator; in this way we have normalized processes that tend to zero (as happens for N ν 1 ,λ 1 (t)−N ν 2 ,λ 2 (t) t and S λ (Lν (t)) t in this paper).In future work one could consider more general models with (independent) random time-changes in terms of an inverse of a general subordinator (see e.g. the recent reference [15]) with the same slowing effect.
The results in this work (and in [3]) concern one-dimensional cases.It could be nice to obtain sample-path versions of these results.One could try to combine the sample-path results for lighttailed Lévy processes in [9] (which can be applied to non-fractional Skellam processes) with possible sample-path results for the inverse of the stable subordinator; unfortunately the derivation of sample-path results for the inverse of the stable subordinator seems to be a difficult task.
We conclude with a discussion on the connection between the scaling exponents of the normalizing factors for the moderate deviations of the fractional Skellam processes, and the concept of long-range dependence (LRD).We recall (see e.g.Definition 2.1 in [16]) that a non-stationary stochastic process {X(t) : t ≥ 0} has the long-range dependence if we have the following property for the correlation coefficient: Cov(X(t), X(s)) Var[X(t)]Var[X(s)] ∼ c(s)t −h (as t → ∞), for some c(s) > 0 and h ∈ (0, 1).Then we have the following statements on the scaling exponents α 1 (ν) in ( 7) and α 2 (ν) in (10).
Funding.CM acknowledges the support of MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (CUP E83C18000100006 and CUP E83C23000330006), of University of Rome Tor Vergata (project "Asymptotic Methods in Probability" (CUP E89C20000680005) and project "Asymptotic Properties in Probability" (CUP E83C22001780005)) and of Indam-GNAMPA.
[4][5]s[4][5].A family of numbers {v t : t > 0} such that v t → ∞ (as t → ∞) is called a speed function, and a lower semicontinuous function I : R → [0, ∞] is called a rate function.Then {Z t : t > 0} satisfies the large deviation principle (LDP from now on) with speed v t and a rate function I if variables as in Remark 2.1, and {N ν,λ 1 +λ 2 (t) : t ≥ 0} is a fractional Poisson process.