86 research outputs found
Large deviations for i.i.d. replications of the total progeny of a Galton--Watson process
The Galton--Watson process is the simplest example of a branching process.
The relationship between the offspring distribution, and, when the extinction
occurs almost surely, the distribution of the total progeny is well known. In
this paper, we illustrate the relationship between these two distributions when
we consider the large deviation rate function (provided by Cram\'{e}r's
theorem) for empirical means of i.i.d. random variables. We also consider the
case with a random initial population. In the final part, we present large
deviation results for sequences of estimators of the offspring mean based on
i.i.d. replications of total progeny.Comment: Published at http://dx.doi.org/10.15559/16-VMSTA72 in the Modern
Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA)
by VTeX (http://www.vtex.lt/
Multivariate fractional Poisson processes and compound sums
In this paper we present multivariate space-time fractional Poisson processes
by considering common random time-changes of a (finite-dimensional) vector of
independent classical (non-fractional) Poisson processes. In some cases we also
consider compound processes. We obtain some equations in terms of some suitable
fractional derivatives and fractional difference operators, which provides the
extension of known equations for the univariate processes.Comment: 19 pages Keywords: conditional independence, Fox-Wright function,
fractional differential equations, random time-chang
Large deviations for fractional Poisson processes
We prove large deviation principles for two versions of fractional Poisson
processes. Firstly we consider the main version which is a renewal process; we
also present large deviation estimates for the ruin probabilities of an
insurance model with constant premium rate, i.i.d. light tail claim sizes, and
a fractional Poisson claim number process. We conclude with the alternative
version where all the random variables are weighted Poisson distributed.
Keywords: Mittag Leffler function; renewal process; random time ch
Asymptotic results for random flights
The random flights are (continuous time) random walkswith finite velocity.
Often, these models describe the stochastic motions arising in biology. In this
paper we study the large time asymptotic behavior of random flights. We prove
the large deviation principle for conditional laws given the number of the
changes of direction, and for the non-conditional laws of some standard random
flights.Comment: 3 figure
Large deviations for risk measures in finite mixture models
Due to their heterogeneity, insurance risks can be properly described as a
mixture of different fixed models, where the weights assigned to each model may
be estimated empirically from a sample of available data. If a risk measure is
evaluated on the estimated mixture instead of the (unknown) true one, then it
is important to investigate the committed error. In this paper we study the
asymptotic behaviour of estimated risk measures, as the data sample size tends
to infinity, in the fashion of large deviations. We obtain large deviation
results by applying the contraction principle, and the rate functions are given
by a suitable variational formula; explicit expressions are available for
mixtures of two models. Finally, our results are applied to the most common
risk measures, namely the quantiles, the Expected Shortfall and the shortfall
risk measures
Random time-change with inverses of multivariate subordinators: governing equations and fractional dynamics
It is well-known that compositions of Markov processes with inverse
subordinators are governed by integro-differential equations of generalized
fractional type. This kind of processes are of wide interest in statistical
physics as they are connected to anomalous diffusions. In this paper we
consider a generalization; more precisely we mean componentwise compositions of
-valued Markov processes with the components of an independent
multivariate inverse subordinator. As a possible application, we present a
model of anomalous diffusion in anisotropic medium, which is obtained as a weak
limit of suitable continuous-time random walks.Comment: 24 page
Correlated fractional counting processes on a finite time interval
We present some correlated fractional counting processes on a finite time
interval. This will be done by considering a slight generalization of the
processes in Borges et al. (2012). The main case concerns a class of space-time
fractional Poisson processes and, when the correlation parameter is equal to
zero, the univariate distributions coincide with the ones of the space-time
fractional Poisson process in Orsingher and Polito (2012). On the other hand,
when we consider the time fractional Poisson process, the multivariate finite
dimensional distributions are different from the ones presented for the renewal
process in Politi et al. (2011). Another case concerns a class of fractional
negative binomial processes
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
Fractional discrete processes: compound and mixed Poisson representations
We consider two fractional versions of a family of nonnegative integer valued
processes. We prove that their probability mass functions solve fractional
Kolmogorov forward equations, and we show the overdispersion of these
processes. As particular examples in this family, we can define fractional
versions of some processes in the literature as the Polya-Aeppli, the Poisson
Inverse Gaussian and the Negative Binomial. We also define and study some more
general fractional versions with two fractional parameters.Comment: 16 pages; 1 figur
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.Comment: 25 page
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