Modern Stochastics: Theory and Applications

The theory of large deviations gives an asymptotic computation of small probabilities on exponential scale (see [3] as a reference of this topic). In particular a large deviation principle (LDP from now on) provides asymptotic bounds for families of probability measures on the same topological space; these bounds are expressed in terms of a speed function (that tends to infinity) and a nonnegative lower semicontinuous rate function defined on the topological space.

The term moderate deviations is used for a class of LDPs which fills the gap between the two following asymptotic regimes: a convergence to a constant ${x_{0}}$ (at least in probability), and a weak convergence to a nonconstant centered Gaussian random variable. The convergence to ${x_{0}}$ is governed by a reference LDP with speed ${v_{t}}$, and a rate function I, say, that uniquely vanishes at ${x_{0}}$ (i.e. $I(x)=0$ if and only if $x={x_{0}}$). Then we have a class of LDPs which depends on the choice of some positive scalings $\{{a_{t}}:t\gt 0\}$ such that ${a_{t}}\to 0$ and ${a_{t}}{v_{t}}\to \infty $ (as $t\to \infty $).

In this paper the topological space mentioned above is the real line $\mathbb{R}$ equipped with the Borel σ-algebra, ${x_{0}}=0\in \mathbb{R}$ and ${v_{t}}=t$; thus we shall have

In some recent papers (see, e.g., [4] and some references cited therein), the term noncentral moderate deviations has been introduced when one has the situation described above, but the weak convergence is towards a nonconstant and non-Gaussian distributed random variable. A multivariate example is given in [10].

A possible way to construct examples of moderate deviation results is the following: $\{{C_{t}}:t\gt 0\}$ is a family of random variables which converges to zero as $t\to \infty $ (and satisfies the reference LDP with a certain rate function and a certain speed ${v_{t}}$); moreover, for a certain function ϕ such that $\phi (x)\to \infty $ as $x\to \infty $, $\{\phi ({v_{t}}){C_{t}}:t\gt 0\}$ converges weakly to some nondegenerating random variable; then, for every family of positive scalings $\{{a_{t}}:t\gt 0\}$ such that ${a_{t}}\to 0$ and ${a_{t}}{v_{t}}\to \infty $ (as $t\to \infty $), one should be able to prove the LDP for $\{\phi ({a_{t}}{v_{t}}){C_{t}}:t\gt 0\}$ with a certain rate function and speed $1/{a_{t}}$. We remark that, according to this approach, one typically has $\phi (x)=\sqrt{x}$ for central moderate deviation results (i.e. for the cases in which the weak convergence is towards a normal distribution); moreover, to give an example with a different situation, we have $\phi (x)=x$ for the noncentral moderate deviation result in [7] (note that r and ${\gamma _{r}}$ in that reference plays the role of t and ${a_{t}}$ in this paper). The results in this paper follow this approach with $\phi (x)={x^{\beta }}$ for some $\beta \in (0,1)$; more precisely $\beta =\alpha (\nu )$ (see (7)) in Section 3 and $\beta ={\alpha _{1}}(\nu )$ (see (12)) in Section 4.

We aim to present some extensions of the recent results presented in [1] and [9]. In particular, we recall that a subordinator is a nondecreasing (real-valued) Lévy process. Throughout this paper we always deal with real-valued light-tailed Lévy processes $\{S(t):t\ge 0\}$ described in Condition 1.1, with an independent random time-change in terms of inverse of stable subordinators.

We recall that, if $\{S(t):t\ge 0\}$ in Condition 1.1 is a Poisson process and $\{{L_{\nu }}(t):t\ge 0\}$ is an independent inverse of a stable subordinator, then the process $\{S({L_{\nu }}(t)):t\ge 0\}$ is a (time) fractional Poisson process (see [11]; see also Section 2.4 in [12] for more general time fractional processes).

The aim of this paper is to provide some extensions of recent noncentral moderate deviation results published in the literature. More precisely we mean: For the first item, we have only one (independent) random time-change for $\{S(t):t\ge 0\}$, and we can specify the results to the fractional Skellam processes of type 2 in [8]. For the second item, we shall assume that $\{S(t):t\ge 0\}$ has bounded variation and is not a subordinator; thus $\{S(t):t\ge 0\}$ can be seen as the difference of two independent nonnull subordinators $\{{S_{1}}(t):t\ge 0\}$ and $\{{S_{2}}(t):t\ge 0\}$ (see Lemma 4.1 in this paper). Then, for the second item, we have two (independent) random time-changes for $\{{S_{1}}(t):t\ge 0\}$ and $\{{S_{2}}(t):t\ge 0\}$, and we can specify the results to the fractional Skellam processes of type 1 in [8].

The outline of the paper is as follows. In Section 2 we recall some preliminaries. The extensions presented above in items 1 and 2 are studied in Sections 3 and 4, respectively. In Section 5 we discuss the possibility to have some generalizations with more general random time-changes, and in particular we present Propositions 5.1 and 5.2 which provide a generalization of the reference LDPs in Propositions 3.1 and 4.1, respectively. In Section 6 we present some comparisons between rate functions, and in particular we follow the same lines of the comparisons in Section 5 in [9]. Finally, motivated by potential applications to other fractional processes in the literature, in Section 7 we discuss the case of the difference of two (independent) tempered stable subordinators.

In this section we recall some preliminaries on large deviations and on the inverse of the stable subordinator, together with the Mittag-Leffler function.

Throughout this section we assume that the following condition holds.

The next Propositions 3.1, 3.2 and 3.3 provide a generalization of Propositions 3.1, 3.2 and 3.3 in [1], respectively, in which $\{S(t):t\ge 0\}$ is a compound Poisson process. We start with the reference LDP for the convergence in probability to zero of $\left\{\frac{S({L_{\nu }}(t))}{t}:t\gt 0\right\}$.

Now we present weak convergence results. In view of these results it is useful to consider the following notation:

Now we present the noncentral moderate deviation results.

Throughout this section we assume that the following condition holds.

We have the following consequence of Condition 4.1.

We can assume that the statement in Lemma 4.1 is known even if we do not have an exact reference for that result (however a statement of this kind appears in the Introduction of [2]). The idea of the proof is the following. If $\Pi (dx)$ is the Lévy measure of a Lévy process with bounded variation, then ${1_{(0,\infty )}}(x)\Pi (dx)$ and ${1_{(-\infty ,0)}}(x)\Pi (dx)$ are again Lévy measures of Lévy processes with bounded variation; thus ${1_{(0,\infty )}}(x)\Pi (dx)$ is the Lévy measure associated to the subordinator $\{{S_{1}}(t):t\ge 0\}$, ${1_{(-\infty ,0)}}(x)\Pi (dx)$ is the Lévy measure associated to the opposite of the subordinator $\{{S_{2}}(t):t\ge 0\}$, and $\{{S_{1}}(t):t\ge 0\}$ and $\{{S_{2}}(t):t\ge 0\}$ are independent.

The next Propositions 4.1, 4.2 and 4.3 provide a generalization of Propositions 3.1, 3.2 and 3.3 in [9], respectively, in which $\{S(t):t\ge 0\}$ is a Skellam process (and therefore $\{{S_{1}}(t):t\ge 0\}$ and $\{{S_{2}}(t):t\ge 0\}$ are two Poisson processes with intensities ${\lambda _{1}}$ and ${\lambda _{2}}$, respectively). We start with the reference LDP for the convergence of $\left\{\frac{{S_{1}}({L_{{\nu _{1}}}^{(1)}}(t))-{S_{2}}({L_{{\nu _{2}}}^{(2)}}(t))}{t}:t\gt 0\right\}$ to zero in probability. In this first result the case ${\nu _{1}}\ne {\nu _{2}}$ is allowed.

From now on we assume that ${\nu _{1}}$ and ${\nu _{2}}$ coincide, and therefore we simply consider the symbol ν, where $\nu ={\nu _{1}}={\nu _{2}}$. Moreover we set

In Sections 3 and 4 we have proved two moderate deviation results, i.e. two collections of three results: a reference LDP (Propositions 3.1 and 4.1), a weak convergence result (Propositions 3.2 and 4.2), and a collection of LDPs which depend on some positive scalings $\{{a_{t}}:t\gt 0\}$ which satisfies condition (1) (Propositions 3.3 and 4.3).

Then one can wonder if it is possible to consider more general time-changes to have a generalization of at least one of these three results in each collection. This question is quite natural because, if one looks at the results presented above, the asymptotic behavior of Mittag-Leffler function (see (2)) seems to play a role in the applications of Gärtner–Ellis theorem (thus in all the results, except those of weak convergence).

In what follows we consider random time-changes which satisfy the following condition.

This condition is a quite natural way if one deals with inverse of heavy-tailed subordinators, indeed it is satisfied by inverse stable subordinators. This will be explained in the following remark.

Then we can present a generalization of the reference LDPs presented in Propositions 3.1 and 4.1. The proofs are very similar to the ones presented for those propositions, and we omit the details.

Finally, we discuss a possible way to generalize Propositions 3.3 and 4.3 when Condition 5.1 holds, together with some possible further conditions. Firstly one should have the analogue of Propositions 3.2 and 4.2; thus one should have the weak convergence of $\left\{\frac{S({L_{f}}(t))}{{t^{1-\alpha (f)}}}:t\gt 0\right\}$ (for some $\alpha (f)\in (0,1)$ which plays the role of $\alpha (\nu )$ in (7)) and $\left\{\frac{{S_{1}}({L_{f}^{(1)}}(t))-{S_{2}}({L_{f}^{(2)}}(t))}{{t^{1-{\alpha _{1}}(f)}}}\right\}$ (for some ${\alpha _{1}}(f)\in (0,1)$ which plays the role of ${\alpha _{1}}(\nu )$ in (12)) to some nondegenerating random variables. In particular, we expect that, as it happens in the previous sections (see eqs. (7) and (12)), $\alpha (f)={\alpha _{1}}(f)$ when the Lévy process $\{S(t):t\ge 0\}$ is not centered (i.e. $m\ne 0$ in Section 3, or ${m_{1}}\ne {m_{2}}$ in Section 4), and $1-\alpha (f)=\frac{1-{\alpha _{1}}(f)}{2}$ when the Lévy process $\{S(t):t\ge 0\}$ is centered.

Then one should be able to handle some suitable limits which follow from suitable applications of the Gärtner–Ellis theorem; more precisely, for every choice of the positive scalings $\{{a_{t}}:t\gt 0\}$ such that (1) holds, we mean for the possible generalization of Proposition 3.3, and for the possible generalization of Proposition 4.3.

In our opinion, we can prove the generalizations of the other results by requiring some other conditions, and Condition 5.1 is not enough. We also point out that, if it is possible to prove these generalizations, the rate functions could not have an explicit expression, and only variational formulas would be available; see ${I_{\mathrm{MD}}}(\cdot ;m)$ in Proposition 3.3 (which can be derived from the variational formula (8)) and ${J_{\mathrm{MD}}}$ in Proposition 4.3 (which can be derived from the variational formula (14)).

In this section $\{S(t):t\ge 0\}$ is a real-valued Lévy process as in Condition 1.1, with bounded variation, and it is not a subordinator; thus we can refer to both Conditions 3.1 and 4.1, and in particular (as stated in Lemma 4.1) we can refer to the nontrivial independent subordinators $\{{S_{1}}(t):t\ge 0\}$ and $\{{S_{2}}(t):t\ge 0\}$ such that $\{S(t):t\ge 0\}$ is distributed as $\{{S_{1}}(t)-{S_{2}}(t):t\ge 0\}$. In particular we can refer to the LDPs in Propositions 3.1 and 4.1, which are governed by the rate functions ${I_{\mathrm{LD}}}$ and ${J_{\mathrm{LD}}}$, and to the classes of LDPs in Propositions 3.3 and 4.3, which are governed by the rate functions ${I_{\mathrm{MD}}}(\cdot ;m)={I_{\mathrm{MD}}}(\cdot ;{m_{1}}-{m_{2}})$ and ${J_{\mathrm{MD}}}$. All these rate functions uniquely vanish at $x=0$. So, by arguing as in [9] (Section 5), we present some generalizations of the comparisons between rate functions (at least around $x=0$) presented in that reference. Those comparisons allow us to compare different convergences to zero; this could be explained by adapting the explanations in [9] (Section 5), and here we omit the details.

We start by comparing ${J_{\mathrm{LD}}}$ in Proposition 4.1 and ${I_{\mathrm{LD}}}$ in Proposition 3.1.

The next Proposition 6.2 provides a similar result which concerns the comparison of ${J_{\mathrm{MD}}}$ in Proposition 4.3 and ${I_{\mathrm{MD}}}(\cdot ;m)={I_{\mathrm{MD}}}(\cdot ;{m_{1}}-{m_{2}})$ in Proposition 3.3. In particular, we have ${I_{\mathrm{MD}}}(x;{m_{1}}-{m_{2}})\gt {J_{\mathrm{MD}}}(x)\gt 0$ for all $x\ne 0$ if and only if ${m_{1}}\ne {m_{2}}$ (note that, if ${m_{1}}\ne {m_{2}}$, we have $m={m_{1}}-{m_{2}}\ne 0$; thus $\alpha (\nu )$ in (7) coincides with ${\alpha _{1}}(\nu )$ in (12)); on the contrary, if ${m_{1}}={m_{2}}$, we have ${I_{\mathrm{MD}}}(x;0)\gt {J_{\mathrm{MD}}}(x)\gt 0$ only if $|x|$ is small enough (and strictly positive).

We remark that the inequalities around $x=0$ in Propositions 6.1 and 6.2 are not surprising; indeed we expect to have a slower convergence to zero when we deal with two independent random time-changes (because in that case we have more randomness).

Now we consider comparisons between rate functions for different values of $\nu \in (0,1)$. We mean the rate functions in Propositions 3.1 and 4.1, and we restrict our attention to a comparison around $x=0$. In view of what follows, we consider some slightly different notation: ${I_{\mathrm{LD},\nu }}$ in place of ${I_{\mathrm{LD}}}$ in Proposition 3.1; ${J_{\mathrm{LD},\nu }}$ in place of ${J_{\mathrm{LD}}}$ in Proposition 4.1, with ${\nu _{1}}={\nu _{2}}=\nu $ for some $\nu \in (0,1)$.

As a consequence of Proposition 6.3 we can say that, in all the above convergences to zero governed by some rate function (that uniquely vanishes at zero), the smaller the ν, the faster the convergence of the random variables to zero.

We also remark that we can obtain a version of Proposition 6.3 in terms of the rate functions ${I_{\mathrm{MD}}}(\cdot ;{m_{1}}-{m_{2}})={I_{\mathrm{MD},\nu }}(\cdot ;{m_{1}}-{m_{2}})$ in Proposition 3.3 and ${J_{\mathrm{MD}}}={J_{\mathrm{MD},\nu }}$ in Proposition 4.3. Indeed we can obtain the same kind of inequalities, and this is easy to check because we have explicit expressions of the rate functions (here we omit the details).

Throughout this section we consider the examples presented below.

Note that, to be consistent in the comparisons between ${I_{\mathrm{LD}}}$ and ${J_{\mathrm{LD}}}$, we always take the rate function ${J_{\mathrm{LD}}}$ in Proposition 4.1 with ${\nu _{1}}={\nu _{2}}=\nu $. Moreover, we remark that, for Example 7.1, the function ${\Lambda _{\nu ,S}}$ is essentially smooth because

indeed these conditions hold if and only if ${\kappa _{S}}({r_{1}})$ and ${\kappa _{S}}(-{r_{2}})$ are positive, and in fact we have

We start with Figure 2. The graphs agree with the inequalities in Proposition 6.3. Moreover Figure 2 is more informative than Figure 3 in [9]; indeed the graphs of ${J_{\mathrm{LD}}}={J_{\mathrm{LD},\nu }}$ at the bottom in Figure 2 (for different values of ν) show that the inequalities in Proposition 6.3 hold only in a neighborhood of the origin $x=0$.

In Figures 3 and 4 we take different values of β and of ${r_{2}}$, respectively, when the other paramaters are fixed. The graphs in these figures agree with Proposition 6.1.