scholarly journals Large deviations for the empirical measure of heavy-tailed Markov renewal processes

2016 ◽  
Vol 48 (3) ◽  
pp. 648-671 ◽  
Author(s):  
Mauro Mariani ◽  
Lorenzo Zambotti
Keyword(s):  
Large Deviations ◽  
Finite Graph ◽  
Rate Function ◽  
Empirical Measure ◽  
Markov Renewal Process ◽  
Large Deviations Principle ◽  
Arrival Times ◽  
Heavy Tailed Distributions ◽  
Markov Renewal Processes ◽  
Heavy Tailed

Abstract A large deviations principle is established for the joint law of the empirical measure and the flow measure of a Markov renewal process on a finite graph. We do not assume any bound on the arrival times, allowing heavy-tailed distributions. In particular, the rate function is in general degenerate (it has a nontrivial set of zeros) and not strictly convex. These features show a behaviour highly different from what one may guess with a heuristic Donsker‒Varadhan analysis of the problem.

scholarly journals Large deviations for randomly connected neural networks: II. State-dependent interactions

2018 ◽  
Vol 50 (3) ◽  
pp. 983-1004 ◽  
Author(s):  
Tanguy Cabana ◽  
Jonathan D. Touboul
Keyword(s):  
Neural Networks ◽  
Large Deviations ◽  
Rate Function ◽  
Network Size ◽  
Kuramoto Model ◽  
Empirical Measure ◽  
Large Deviations Principle ◽  
Stochastic Version ◽  
State Dependent ◽  
Spatially Extended

Abstract We continue the analysis of large deviations for randomly connected neural networks used as models of the brain. The originality of the model relies on the fact that the directed impact of one particle onto another depends on the state of both particles, and they have random Gaussian amplitude with mean and variance scaling as the inverse of the network size. Similarly to the spatially extended case (see Cabana and Touboul (2018)), we show that under sufficient regularity assumptions, the empirical measure satisfies a large deviations principle with a good rate function achieving its minimum at a unique probability measure, implying, in particular, its convergence in both averaged and quenched cases, as well as a propagation of a chaos property (in the averaged case only). The class of model we consider notably includes a stochastic version of the Kuramoto model with random connections.

scholarly journals Large deviations for randomly connected neural networks: I. Spatially extended systems

2018 ◽  
Vol 50 (3) ◽  
pp. 944-982 ◽  
Author(s):  
Tanguy Cabana ◽  
Jonathan D. Touboul
Keyword(s):  
Neural Networks ◽  
Large Deviations ◽  
Rate Function ◽  
Network Size ◽  
Empirical Measure ◽  
Interaction Coefficients ◽  
Large Deviations Principle ◽  
Implicit Equation ◽  
Random Interaction ◽  
Spatially Extended

Abstract In a series of two papers, we investigate the large deviations and asymptotic behavior of stochastic models of brain neural networks with random interaction coefficients. In this first paper, we take into account the spatial structure of the brain and consider first the presence of interaction delays that depend on the distance between cells and then the Gaussian random interaction amplitude with a mean and variance that depend on the position of the neurons and scale as the inverse of the network size. We show that the empirical measure satisfies a large deviations principle with a good rate function reaching its minimum at a unique spatially extended probability measure. This result implies an averaged convergence of the empirical measure and a propagation of chaos. The limit is characterized through a complex non-Markovian implicit equation in which the network interaction term is replaced by a nonlocal Gaussian process with a mean and covariance that depend on the statistics of the solution over the whole neural field.

scholarly journals Run-and-Tumble Motion: The Role of Reversibility

2021 ◽  
Vol 183 (3) ◽  
Author(s):  
Bart van Ginkel ◽  
Bart van Gisbergen ◽  
Frank Redig
Keyword(s):  
Free Energy ◽  
Diffusion Coefficient ◽  
Large Deviations ◽  
Energy Function ◽  
Internal State ◽  
Rate Function ◽  
Free Energy Function ◽  
Large Deviations Principle ◽  
Preferred Direction ◽  
State Process

AbstractWe study a model of active particles that perform a simple random walk and on top of that have a preferred direction determined by an internal state which is modelled by a stationary Markov process. First we calculate the limiting diffusion coefficient. Then we show that the ‘active part’ of the diffusion coefficient is in some sense maximal for reversible state processes. Further, we obtain a large deviations principle for the active particle in terms of the large deviations rate function of the empirical process corresponding to the state process. Again we show that the rate function and free energy function are (pointwise) optimal for reversible state processes. Finally, we show that in the case with two states, the Fourier–Laplace transform of the distribution, the moment generating function and the free energy function can be computed explicitly. Along the way we provide several examples.

scholarly journals Large Deviations for a Class of Multivariate Heavy-Tailed Risk Processes Used in Insurance and Finance

2021 ◽  
Vol 14 (5) ◽  
pp. 202
Author(s):  
Miriam Hägele ◽  
Jaakko Lehtomaa
Keyword(s):  
Large Deviations ◽  
Risk Sharing ◽  
Dependence Structure ◽  
Large Deviations Principle ◽  
Shape Constraint ◽  
Multiple Components ◽  
Insurance Portfolio ◽  
Heavy Tailed ◽  
Tail Events ◽  
Risk Processes

Modern risk modelling approaches deal with vectors of multiple components. The components could be, for example, returns of financial instruments or losses within an insurance portfolio concerning different lines of business. One of the main problems is to decide if there is any type of dependence between the components of the vector and, if so, what type of dependence structure should be used for accurate modelling. We study a class of heavy-tailed multivariate random vectors under a non-parametric shape constraint on the tail decay rate. This class contains, for instance, elliptical distributions whose tail is in the intermediate heavy-tailed regime, which includes Weibull and lognormal type tails. The study derives asymptotic approximations for tail events of random walks. Consequently, a full large deviations principle is obtained under, essentially, minimal assumptions. As an application, an optimisation method for a large class of Quota Share (QS) risk sharing schemes used in insurance and finance is obtained.

scholarly journals Large Deviations for the Single-Server Queue and the Reneging Paradox

2021 ◽  
Author(s):  
Rami Atar ◽  
Amarjit Budhiraja ◽  
Paul Dupuis ◽  
Ruoyu Wu
Keyword(s):  
Large Deviations ◽  
Decay Rate ◽  
Sample Path ◽  
Arrival Rate ◽  
Rate Function ◽  
Service Rate ◽  
Single Server ◽  
Large Deviations Principle ◽  
Long Run ◽  
The Individual

For the M/M/1+M model at the law-of-large-numbers scale, the long-run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: the decay rate of the probability that the long-run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.

Large deviations for the time of ruin

1999 ◽  
Vol 36 (3) ◽  
pp. 733-746 ◽  
Author(s):  
Harri Nyrhinen
Keyword(s):  
Large Deviations ◽  
Positive Real Number ◽  
Rate Function ◽  
Positive Real ◽  
Large Deviations Principle ◽  
Borel Sets ◽  
Time Of Ruin ◽  
The Family ◽  
The Time Of Ruin ◽  
Simulation Problem

Let {Yn | n=1,2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Yn > M} (T = +∞ if Yn ≤ M for n=1,2,…). We are interested in the ruin probabilities for large M. Define the family of measures {PM | M > 0} by PM(B) = P(T/M ∊ B) for B ∊ ℬ (ℬ = Borel sets of ℝ). We prove that for a wide class of processes {Yn}, the family {PM} satisfies a large deviations principle. The rate function will correspond to the approximation P(T/M ≈ x) ≈ P(Y⌈xM⌉/M ≈ 1) for x > 0. We apply the result to a simulation problem.

scholarly journals Sample Path Large Deviations of Poisson Shot Noise with Heavy-Tailed Semiexponential Distributions

2011 ◽  
Vol 48 (03) ◽  
pp. 688-698 ◽  
Author(s):  
Ken R. Duffy ◽  
Giovanni Luca Torrisi
Keyword(s):  
Large Deviations ◽  
Shot Noise ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Sample Path ◽  
Rate Function ◽  
Sample Paths ◽  
Heavy Tailed ◽  
Poisson Shot Noise ◽  
Sample Path Large Deviations

It is shown that the sample paths of Poisson shot noise with heavy-tailed semiexponential distributions satisfy a large deviation principle with a rate function that is insensitive to the shot shape. This demonstrates that, on the scale of large deviations, paths to rare events do not depend on the shot shape.

scholarly journals Large deviations of means of heavy-tailed random variables with finite moments of all orders

2017 ◽  
Vol 54 (1) ◽  
pp. 66-81
Author(s):  
Jaakko Lehtomaa
Keyword(s):  
Large Deviations ◽  
Hazard Function ◽  
The Novel ◽  
Exponential Distributions ◽  
Large Deviations Principle ◽  
Novel Approach ◽  
Logarithmic Asymptotics ◽  
The Mean ◽  
Heavy Tailed ◽  
Finite Moments

AbstractLogarithmic asymptotics of the mean process {Sn∕n} are investigated in the presence of heavy-tailed increments. As a consequence, a full large deviations principle for means is obtained when the hazard function of an increment is regularly varying with index α∈(0,1). This class includes all stretched exponential distributions. Thus, the previous research of Gantert et al. (2014) is extended. Furthermore, the presented proofs are more transparent than the techniques used by Nagaev (1979). In addition, the novel approach is compatible with other common classes of distributions, e.g. those of lognormal type.

scholarly journals Heavy-tailed distributions in a stochastic gene autoregulation model

2021 ◽  
Author(s):  
Pavol Bokes
Keyword(s):  
Large Deviations ◽  
Stationary Distribution ◽  
Burst Size ◽  
Cumulative Effect ◽  
Jump Process ◽  
Gene Products ◽  
Tail Region ◽  
Expression Variability ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed

Synthesis of gene products in bursts of multiple molecular copies is an important source of gene expression variability. This paper studies large deviations in a Markovian drift--jump process that combines exponentially distributed bursts with deterministic degradation. Large deviations occur as a cumulative effect of many bursts (as in diffusion) or, if the model includes negative feedback in burst size, in a single big jump. The latter possibility requires a modification in the WKB solution in the tail region. The main result of the paper is the construction, via a modified WKB scheme, of matched asymptotic approximations to the stationary distribution of the drift--jump process. The stationary distribution possesses a heavier tail than predicted by a routine application of the scheme.

scholarly journals Multipopulation Spin Models: A View from Large Deviations Theoretic Window

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Alex Akwasi Opoku ◽  
Godwin Osabutey
Keyword(s):  
Large Deviations ◽  
Probability Distributions ◽  
Mean Field ◽  
Variational Formula ◽  
Rate Function ◽  
Large Deviations Principle ◽  
Spin Models ◽  
Empirical Measures ◽  
Rate Functions ◽  
Empirical Means

This paper studies large deviations properties of vectors of empirical means and measures generated as follows. Consider a sequence X1,X2,…,Xn of independent and identically distributed random variables partitioned into d-subgroups with sizes n1,…,nd. Further, consider a d-dimensional vector mn whose coordinates are made up of the empirical means of the subgroups. We prove the following. (1) The sequence of vector of empirical means mn satisfies large deviations principle with rate n and rate function I, when the sequence X1,X2,…,Xn is Rl valued, with l≥1. (2) Similar large deviations results hold for the corresponding sequence of vector of empirical measures Ln if Xi’s, i=1,2,…,n, take on finitely many values. (3) The rate functions for the above large deviations principles are convex combinations of the corresponding rate functions arising from the large deviations principles of the coordinates of mn and Ln. The probability distributions used in the convex combinations are given by α=(α1,…,αd)=limn→∞1/n(n1,…,nd). These results are consequently used to derive variational formula for the thermodynamic limit for the pressure of multipopulation Curie-Weiss (I. Gallo and P. Contucci (2008), and I. Gallo (2009)) and mean-field Pott’s models, via a version of Varadhan’s integral lemma for an equicontinuous family of functions. These multipopulation models serve as a paradigm for decision-making context where social interaction and other socioeconomic attributes of individuals play a crucial role.

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