The term

The theory of large deviations gives an asymptotic computation of small probabilities on exponential scale; see [

The term

We have the following remarks.

We can recover the asymptotic regimes

Concerning the random variables

Now we present a prototype example of the framework in Assertion

We set

The aim of this paper is to present some examples of

The terminology of noncentral moderate deviations appears in three recent results: Proposition 3.3 in [

The common line of the examples studied in this paper can be summarized as follows.

Some interesting common features of the examples studied in this paper are given by the following equalities:

Now we present the outline of the paper with a very brief description of the examples studied in each section. We start with some preliminaries in Section

We conclude with some notation used throughout the paper. We write

We start with the definition of large deviation principle (see, e.g., [

The following Lemma

It is known that the final statement (

In this section we consider the following example.

Let

Throughout this section we set

Thus we are studying a sequence of random variables that has interest for any possible concrete model related to the maximum domain of attraction of the Weibull distribution (indeed, exponential distribution is a particular case of the Weibull distribution). For instance, the random variables

Now we prove the moderate deviation result.

We apply Lemma

We have to show that, for every

We have to show that, for every

We want to show that, for every

In this section we consider the following example.

Let

Here we list some particular cases in which

Standard normal distribution (see, e.g., [

Gamma distribution for

Weibull distribution for

Logistic distribution

We need several consequences of the assumptions in Example

It is known (see, e.g., [

Recall from (

If we consider the four distributions listed above for which the function

Some further preliminaries are needed. Firstly, under the assumptions in Example

By the well-known Karamata’s representation of slowly varying functions (see, e.g., Theorem 1.3.1 in [

Firstly we recall that

We have (make the change of variable

We have (make the change of variable

Throughout this section we set

Thus we are studying a sequence of random variables that has interest for any possible concrete model related to the maximum domain of attraction of the Gumbel distribution. A possible example is when the random variables

Now we prove the moderate deviation result. We shall see that, for this example, the rate functions

We apply Lemma

We have to show that, for every

We have to show that, for every

We want to show that, for every

In this section we consider the following example.

Let

It is well known that the random variable

Now we prove the moderate deviation result. We shall see that, for this example, the rate functions

We apply Lemma

We have to show that, for every

We have to show that, for every

We want to show that, for every

We conclude the proof of (

Concerning

We remark that

Now, by using the same computations as in the proof of Proposition 2.1 in [

In this section we consider the following example.

Let

Note that, for every

In general, the distribution functions of the random variables

This example does not seem to have interesting applications to concrete models. However it could give some interesting ideas for the construction of more advanced examples based on other replacement models (see, e.g., the replacement model in [

Throughout this section we set

Now we prove the LDP concerning the convergence to zero in

We apply Lemma

For every

For every

We want to show that, for every

If

If

Now we prove the moderate deviation result. We shall see that, for this example, the family of positive numbers

We apply Lemma

For every

For every

We show that, for every

If

If

We thank the referees for their comments.