We consider a sequence of fractional Ornstein–Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with a kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.

Recently, there has been a certain interest in the literature about the convergence of the

The construction of this kind kind of processes was introduced in [

The

In order to extend the above construction, we recall the notion of stochastic Volterra processes defined as convolution processes between a given kernel and a Brownian motion

These processes cannot be expressed in the Itô differential form (

In analogy with the existing literature, we aim to construct a class of stochastic Volterra processes via an aggregation procedure (i.e. taking the limit of the empirical means). Thanks to this approach, we are able to prove some of its properties. We shall refer to our processes as

In this paper, we assume that

Notice that condition (

Finally, in Section

We show that the process

We introduce the Mittag-Leffler function

The series in (

The case

The graph of the Mittag-Leffler function

The global behaviour of the Mittag-Leffler function

Recall that, by definition, the two-parametric Mittag-Leffler function is (see, for example, [

In this paper, we will encounter the following special function,

In the next lemma, we prove that

Since

By using again Stirling’s approximation, it is possible to compute the

Further properties of the function

The graph of the function

We consider in this section the solution of the fractional evolution equation (

The relation between Mittag-Leffler function and Riemann–Liouville fractional derivative is specified through the following relation [

For the sake of completeness, let us recall the behaviour of the Gamma distribution.

We apply this proposition to compute the expected value (with respect to the measure

Let us consider the representation of the scalar resolvent kernel

In this section, we will study the asymptotic behaviour of the empirical mean process

In order to prove the convergence of

Moreover, the function

We have proved before that

In the remaining part of this section, the whole construction is thought to hold

By [

convergence of the finite dimensional distributions, and

the moment condition

We shall prove each of the above-described conditions in the following two lemmas, which therefore together imply the thesis of the theorem.

Both

First, we notice that by exploiting the Gaussianity of

In conclusion, we get

In this section, we consider the convergence of the paths of

Our proof will follow from a somehow broader result. Let

We are now ready to prove the main result of this section.

With the notation

Therefore, we get

By another application of the law of large numbers we get that the sequence

Finally, since

In this section we discuss the asymptotic behaviour as

A standard assumption in this section will be the following:

By a direct computation, we see that the random variable

We first extend the Wiener process

Now, for any

In next lemma we prove that

Recall that

We estimate

The result follows from the previous lemma. We have seen that

Now, a simple modification of the computations leading to Lemma

Let us consider the following modification of our setting. Let

The only point worth some care is the fact that the variance is finite. This follows from the computation in [

This follows from the computation leading to the proof of Lemma

Before we provide the proof, let us recall a few known results about convergence of a sequence of Gaussian random variables (see, e.g., [

By definition, the difference

As opposed to Theorem

By setting

It remains to prove a dominated convergence theorem which holds

Therefore, outside an event of zero measure we have, eventually in

In order to conclude the proof, it remains to apply the dominated convergence theorem to the integrals in (

The next, and final, step is to prove that the weak convergence of the sequence of processes

Notice that the proof is quite similar to that of Theorem

The convergence of the finite dimensional distributions follows from the fact that all the processes involved are Gaussian and thus we have established

The next result is the analogue of Lemma

By an analysis of the proof of Lemma

We would like to thank Prof. Luisa Beghin for a fruitful discussion and for pointing us towards Wright–Fox generalized functions, as well as the anonymous referees for their comments and remarks, which helped us in improving the paper.