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Rita Giuliano

Publications and source records attributed to Rita Giuliano.

16 recordsLinked to original sources
arXiv2026

A large deviation principle for a class of weighted means of random variables which converges weakly to the Dickman distribution

In this paper we consider a wide class of weighted means of random variables which converge weakly to the Dickman distribution. This is inspired by a result proved in [3]. Then we prove a large deviation principle for the sequence of these weighted means. In particular we recover a result proved in [8]. Moreover the generalized framework of this paper allows to consider suitable Neyman Type A distributed random variables.

math.PR↗
arXiv2024

On the limit distribution of extremes of generalized Oppenheim random variables

This paper investigates the asymptotic behavior of the extremes of a sequence of generalized Oppenheim random variables. Particularly, we establish conditions under which some normalized extremes of sequences arising from Oppenheim expansions belong to the maximum domain of attraction of the Frechet distribution. Additionally, we identify conditions under which the maxima and minima of Oppenheim random variables demonstrate some kind of asymptotic independence. Finally, we prove an Extreme Types theorem for Oppenheim expansions with unknown dependent structure.

math.PR↗
arXiv2024

The Beurling-Malliavin density, the Polya density and their connection

In this paper we present a new formulation of the Beurling-Malliavin density (Proposition 1). Then we consider the upper Polya density and show how its existence is connected with the concept of subadditivity; moreover, by means of some quantities introduced for proving Proposition 1, a theorem is presented that clarifies the connection between the upper Polya and the Beurling-Malliavin densities. In the last section we discuss the classical definition of the upper Polya density and we prove a result which seems to be new.

math.NT↗
arXiv2023

On the connection between the Beurling-Malliavin density and the asymptotic density

We study the notion of Beurling-Malliavin density from the point of view of Number Theory. We prove a general relation between the Beurling-Malliavin density and the upper asymptotic density; we identify a class of sequences for which the two densities coincide; this class contains the arithmetic progressions. Last, by means of an alternative definition of Beurling-Malliavin density, we study the connection with the asymptotic density for another kind of sequences that again generalizes the arithmentic progressions.

math.NT↗
arXiv2023

Strong laws of large numbers for lightly trimmed sums of generalized Oppenheim expansions

In the framework of generalized Oppenheim expansions we prove strong law of large numbers for lightly trimmed sums. In the first part of this work we identify a particular class of expansions for which we provide a convergence result assuming that only the largest summand is deleted from the sum; this result generalizes a strong law recently proven for the Luroth case. In the second part we drop any assumptions concerning the structure of the Oppenheim expansions and we prove a result concerning trimmed sums when at least two summands are trimmed; then we derive a corollary for the case in which only the largest summand is deleted from the sum.

math.PR↗
arXiv2023

Intermediately Trimmed Sums of Oppenheim Expansions: a Strong Law

The work of this paper is devoted to obtaining strong laws for intermediately trimmed sums of random variables with infinite means. Particularly, we provide conditions under which the intermediately trimmed sums of independent but not identically distributed random variables converge almost surely. Moreover, by dropping the assumption of independence we provide a corresponding convergence result for a special class of Oppenheim expansions. We highlight that the results of this paper generalize the results provided in the recent work of \cite{KS} while the convergence of intermediately trimmed sums of generalized Oppenheim expansions is studied for the first time.

math.PR↗
arXiv2023

Asymptotic results for sums and extremes

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 a convergence in probability of some random variables to a constant and a weak convergence to a centered Gaussian distribution (when such random variables are properly centered and rescaled). We talk about noncentral moderate deviations when the weak convergence is towards a non-Gaussian distribution. In this paper, we prove a noncentral moderate deviation result for the bivariate sequence of sums and maxima of i.i.d. random variables bounded from above. We also prove a result where the random variables are not bounded from above, and the maxima are suitably normalized. Finally, we prove a moderate deviation result for sums of partial minima of i.i.d. exponential random variables.

math.PR↗
arXiv2022

New asymptotic results for generalized Oppenheim expansions

In this work, we study convergence in probability and almost sure convergence for weighted partial sums of random variables that are related to the class of generalized Oppenheim expansions. It is worth noting that the random variables under study have infinite mean and the results are obtained without any dependence assumptions.

math.PR↗
arXiv2022

Some examples of non-central moderate deviations for sequences of real random variables

The term \emph{moderate deviations} is often used in the literature to mean a class of large deviation principles that, in some sense, fill the gap between a convergence in probability to zero (governed by a large deviation principle) and a weak convergence to a centered Normal distribution. In this paper we present some examples of classes of large deviation principles of this kind, but the involved random variables converge weakly to Gumbel, exponential and Laplace distributions.

math.PR↗
arXiv2021

Asymptotics of running maxima for $φ$-subgaussian random double arrays

The article studies the running maxima $Y_{m,j}=\max_{1 \le k \le m, 1 \le n \le j} X_{k,n} - a_{m,j}$ where $\{X_{k,n}, k \ge 1, n \ge 1\}$ is a double array of $φ$-subgaussian random variables and $\{a_{m,j}, m\ge 1, j\ge 1\}$ is a double array of constants. Asymptotics of the maxima of the double arrays of positive and negative parts of $\{Y_{m,j}, m \ge 1, j \ge 1\}$ are studied, when $\{X_{k,n}, k \ge 1, n \ge 1\}$ have suitable "exponential-type" tail distributions. The main results are specified for various important particular scenarios and classes of $φ$-subgaussian random variables.

math.PR↗
arXiv2020

Convergence for weighted sums of Luroth type random variables

In this work we prove an asymptotic result, that under some conditions on the involved distribution functions, is valid for any Oppenheim expansion, extending a classical result proven by W. Vervaat in 1972 for denominators of the Luroth case. Furthermore, we study the convergence in distribution of weighted sums of a sequence of independent random variables. Although the result is of its own interest, in the present setting it is used to prove convergence in distribution of specific sequences of random variables generalizing known results obtained for Luroth random variables.

math.PR↗
arXiv2020

On exact laws of large numbers for Oppenheim expansions with infinite mean

In this work we investigate the asymptotic behaviour of weighted partial sums of a particular class of random variables related to Oppenheim series expansions. More precisely, we verify convergence in probability as well as almost sure convergence to a strictly positive and finite constant without assuming any dependence structure or the existence of means. Results of this kind are known as exact weak and exact strong laws.

math.PR↗
arXiv2015

Local Limit Theorems in some Random models from Number Theory

We study the local limit theorem for weighted sums of Bernoulli variables. We show on examples that this is an important question in the general theory of the local limit theorem, and which turns up to be not well explored. The examples we consider arise from standard random models used in arithmetical number theory. We next use the characteristic function method to prove new local limit theorems for weighted sums of Bernoulli variables. Further, we give an application of the almost sure local limit theorem to a representation problem in additive number theory due to Burr, using an appropriate random model. We also give a simple example showing that the local limit theorem, in its standard form, fails to be sharp enough for estimating the probability $P\{S_n\in E\}$ for infinite sets of integers $E$, already in the simple case where $S_n$ is a sum of $n$ independent standard Bernoulli random variables and $E$ an arithmetic progression.

math.PR↗
arXiv2014

Approximate Local Limit Theorems with Effective Rate and Application to Random Walks in Random Scenery

We show that the Bernoulli part extraction method can be used to obtain approximate forms of the local limit theorem for sums of independent lattice valued random variables, with effective error term, that is with explicit parameters and universal constants. We also show that our estimates allow to recover Gnedenko and Gamkrelidze local limit theorems. We further establish by this method a local limit theorem with effective remainder for random walks in random scenery.

math.PR↗
arXiv2013

A general correlation inequality and the Almost Sure Local Limit Theorem for random sequences in the domain of attraction of a stable law

In the present paper we obtain a new correlation inequality and use it for the purpose of extending the theory of the Almost Sure Local Limit Theorem to the case of lattice random sequences in the domain of attraction of a stable law. In particular, we prove ASLLT in the case of the normal domain of attraction of $α$--stable law, $α\in(1,2)$.

math.PR↗