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

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

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 $\mathbb{R}^d$-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

Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation

In the renewal risk model, we study the asymptotic behavior of the expected time-integrated negative part of the process. This risk measure has been introduced by Loisel (2005). Both heavy-tailed and light-tailed claim amount distributions are investigated. The time horizon may be finite or infinite. We apply the results to an optimal allocation problem with two lines of business of an insurance company. The asymptotic behavior of the two optimal initial reserves are computed.Ruin theory; heavy-tailed and light-tailed claim size distribution; risk measure; optimal reserve allocation

Structural and mechanical feasibility study of a variable camber wing (VCW) for a transport aircraft

Aerodynamic investigations have shown' that variable camber wings (VCW) for transport aircraft have considerable potential in terms of improving aircraft performance and enhancing their operational flexibility. In order to justify these benefits it is essential that the camber varying system is structurally and mechanically feasible. This research examined the feasibility of providing variable camber to two supercritical aerofoil sections of different'characteristics. The unique method of camber vaTiation was applied by rotating the forward and aft regions of the aerofoil on a circular arc and keeping the surface continuous and matching at their attachment to the main wing box. The change in camber thus increased the chord due to translational motion of the aforementioned regions. The geometries required for varying the forward camber by this method presented formidable design difficulties and no immediate solutions could be found. As a result, an alternative geometry was devised which accepts camber by simply drooping the nose region. A novel idea was developed for aft camber variation, which is considered to be universal for all supercritical aerofoil sections. The system utilises a tracking mechanism which guides a trailing edge element on a continuous arc. Surface continuity is provided by a flexible skin on the upper side and a spring loaded hinged panel on the under side. The flexible skin remains attached to the trailing edge element through a series of roller link arrangement which locate the skin in a separate guide rail. The large moment arm and therefore the increased torsional loads created due to the translational motion of the trailing edge element necessitated investigation of alternative deployment geometries. As a result two additional geometries were schemed. One had reduced radius of rotation and therefore reduced extension, while the other changed camber by drooping the aft region without any chordal extension. Since there was no aerodynamic evidence on the possible benefits offered by these geometries it was decide to postpone them until such information was available. Some detailed aspects of the proposed concept for aft camber variation were considered by applying the system to a modem transport aircraft wing. This resulted in a design which is practically feasible. Justification of this concept was made by designing and testing a half scale structural model of one trailing edge segment. Three dimensional (3-D) geometric investigation showed that the camber-varying elements ride on a frustum of a cone and therefore their deployment is skewed to the line of flight. The 3-D geometric implications of variable camber clearly suggested that the camber variation by rotation on a circular arc, on a tapered wing can be possible if the rotating element is made to flex and twist or it utilises a pin jointed arrangement. To provide the necessary flexibility to the trailing edge element, its structural box best be made from fibre reinforced plastic material. The deployment of the trailing edge element on the structural m(; del was made possible by designing it in laminated wood. Comparison of the proposed variable camber system with a conventional single slotted flap arrangement suggests that the two systems could be equally complex but the variable camber could be slightly heavier., Further systems investigations are required to quantify overall aerodynamic, mass, and cost implications of the use of VCW on transport aircraft