arXiv subjects

Claudio Macci

Publications and source records attributed to Claudio Macci.

At least 19 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↗

arXiv2026

Analysis of multivariate symbol statistics in primitive rational models

We study the asymptotic behaviour of sequences of multivariate random variables representing the number of occurrences of a given set of symbols in a word of length $n$ generated at random according to a rational stochastic model. Assuming primitive the matrix of the total weights of transitions of the model, we first determine asymptotic expressions for the mean values and the covariances of such statistics. Then we establish two asymptotic results that generalize known univariate cases to different regimes: a large deviation principle with speed $n$, implying almost sure convergence, and a multivariate Gaussian limit. Additionally, we introduce a novel moderate deviation result as a bridge between these regimes. Central to our proofs is a quasi-power property for the moment generating function of the statistics, allowing us to employ the G\"artner-Ellis Theorem for both large and moderate deviations.

math.PR↗

arXiv2026

Functional Large Deviations for Wide Deep Neural Networks with Gaussian Initialization and Lipschitz Activations

We establish a functional large deviation principle for fully connected multi-layer perceptrons with i.i.d. Gaussian weights (LeCun initialization) and general Lipschitz activation functions, including therefore the popular case of ReLU. The large deviation principle holds for the entire network output process on any compact input set. The proof combines exponential tightness for recursively defined processes, finite-dimensional large deviations, and the Dawson-G\"artner theorem, extending existing results beyond finite input sets and less general activations.

math.PR↗

arXiv2026

Large deviation principles and functional limit theorems in the deep limit of wide random neural networks

This paper studies large deviation principles and weak convergence, both at the level of finite-dimensional distributions and in functional form, for a class of continuous, isotropic, centered Gaussian random fields defined on the unit sphere. The covariance functions of these fields evolve recursively through a nonlinear map induced by an activation function, reflecting the statistical dynamics of infinitely wide random neural networks as depth increases. We consider two types of centered fields, obtained by subtracting either the value at the North Pole or the spherical average. According to the behavior of the derivative at $t=1$ of the associated covariance function, we identify three regimes: low disorder, sparse, and high disorder. In the low-disorder regime, we establish functional large deviation principles and weak convergence results. In the sparse regime, we obtain large deviation principles and weak convergence for finite-dimensional distributions, while both properties fail at the functional level sense due to the emergence of discontinuities in the covariance recursion.

math.PR↗

arXiv2025

Some vector-valued examples of noncentral moderate deviation results

The term noncentral moderate deviations is used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between the convergence in probability to a constant (governed by a reference large deviation principle) and a weak convergence to a non-Gaussian (and non-degenerating) distribution. Several examples can be found in the literature, mainly for real-valued random variables (see, e.g.,~\cite{GiulianoMacci} and the references cited therein). In this paper we present some examples with vector-valued random variables.

math.PR↗

arXiv2025

Noncentral moderate deviations for time-changed multivariate L\'evy processes with linear combinations of inverse stable subordinators

The term noncentral moderate deviations is used in the literature to mean a class of large deviation principles that, in some sense, fills the gap between the convergence in probability to a constant (governed by a reference large deviation principle) and a weak convergence to a non-Gaussian (and non-degenerating) distribution. Some noncentral moderate deviation results in the literature concern time-changed univariate L\'evy processes, where the time-changes are given by inverse stable subordinators. In this paper we present analogue results for multivariate L\'evy processes; in particular the random time-changes are suitable linear combinations of independent inverse stable subordinators.

math.PR↗

arXiv2024

Large and moderate deviations for Gaussian neural networks

We prove large and moderate deviations for the output of Gaussian fully connected neural networks. The main achievements concern deep neural networks (i.e., when the model has more than one hidden layer) and hold for bounded and continuous pre-activation functions. However, for deep neural networks fed by a single input, we have results even if the pre-activation is ReLU. When the network is shallow (i.e., there is exactly one hidden layer) the large and moderate principles hold for quite general pre-activation functions.

math.PR↗

arXiv2024

Noncentral moderate deviations for time-changed L\'evy processes with inverse of stable subordinators

In this paper we present some extensions of recent noncentral moderate deviation results in the literature. In the first part we generalize the results in \cite{BeghinMacciSPL2022} by considering a general L\'evy process $\{S(t):t\geq 0\}$ instead of a compound Poisson process. In the second part we assume that $\{S(t):t\geq 0\}$ has bounded variation and is not a subordinator; thus $\{S(t):t\geq 0\}$ can be seen as the difference of two independent non-null subordinators. In this way we generalize the results in \cite{LeeMacci} for Skellam processes.

math.PR↗

arXiv2023

Asymptotic results for compound sums in separable Banach spaces

We prove large and moderate deviation results for sequences of compound sums, where the summands are i.i.d. random variables taking values in a separable Banach space. We establish that the results hold by proving that we are dealing with exponentially tight sequences. We present two moderate deviation results: in the first one the summands are centered, in the second one the compound sums are centered.

math.PR↗

arXiv2022

Asymptotic results for the absorption time of telegraph processes with a non-standard barrier at the origin

A telegraph process with an elastic barrier at the origin was studied in [5]; in particular the number of visits of the origin before the absorption is a geometric distributed random variable M. Some asymptotic results (large and moderate deviations) for that model were obtained in [17]. In this paper we study large and moderate deviations for a generalized model where M is a light-tailed distributed random variable.

math.PR↗

arXiv2022

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

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.

math.PR↗

arXiv2022

Non-Universal Moderate Deviation Principle for the Nodal Length of Arithmetic Random Waves

Inspired by the recent work [MRT21], we prove a non-universal non-central Moderate Deviation principle for the nodal length of arithmetic random waves (Gaussian Laplace eigenfunctions on the standard flat torus) both on the whole manifold and on shrinking toral domains. Second order fluctuations for the latter were established in [MPRW16] and [BMW20] respectively, by means of chaotic expansions, number theoretical estimates and full correlation phenomena. Our proof is simple and relies on the interplay between the long memory behavior of arithmetic random waves and the chaotic expansion of the nodal length, as well as on well-known techniques in Large Deviation theory (the contraction principle and the concept of exponential equivalence).

math.PR↗

arXiv2022

Non-central moderate deviations for compound fractional Poisson processes

The term "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. We talk about "non-central moderate deviations" when the weak convergence is towards a non-Gaussian distribution. In this paper we study non-central moderate deviations for compound fractional Poisson processes with light-tailed jumps.

math.PR↗

arXiv2021

Asymptotic results for families of power series distributions

In this paper we consider suitable families of power series distributed random variables, and we study their asymptotic behavior in the fashion of large (and moderate) deviations. We also present two examples of fractional counting processes, where the normalizations of the involved power series distributions can be expressed in terms of the Prabhakar function. The first example allows to consider the counting process in \cite{PoganyTomovski}, the second one is inspired by a model studied in \cite{GarraOrsingherPolito}.

math.PR↗

arXiv2021

An inverse Sanov theorem for exponential families

We prove the large deviation principle (LDP) for posterior distributions arising from subfamilies of full exponential families, allowing misspecification of the model. Moreover, motivated by the so-called inverse Sanov Theorem (see e.g. Ganesh and O'Connell 1999 and 2000), we prove the LDP for the corresponding maximum likelihood estimator, and we study the relationship between rate functions. In our setting, even in the non misspecified case, it is not true in general that the rate functions for posterior distributions and for maximum likelihood estimators are Kullback-Leibler divergences with exchanged arguments.

math.ST↗

arXiv2021

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

Asymptotic results for linear combinations of spacings generated by i.i.d. exponential random variables

We prove large (and moderate) deviations for a class of linear combinations of spacings generated by i.i.d. exponentially distributed random variables. We allow a wide class of coefficients which can be expressed in terms of continuous functions defined on [0, 1] which satisfy some suitable conditions. In this way we generalize some recent results by Giuliano et al. (2015) which concern the empirical cumulative entropies defined in Di Crescenzo and Longobardi (2009a).

math.PR↗