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Large deviations for i.i.d. replications of the total progeny of a Galton–Watson process

Claudio Macci Barbara Pacchiarotti

https://doi.org/10.15559/16-VMSTA72

Pub. online: 11 Jan 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 1–13

Abstract

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 Cramér’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.

2010 Mathematics Subject Classification index

https://doi.org/10.15559/16-VMSTA34MI

Pub. online: 4 Jan 2017 Type: 2010 Mathematics Subject Classification Index

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), pp. 369–372

Subject index

https://doi.org/10.15559/16-VMSTA34SI

Pub. online: 4 Jan 2017 Type: Subject Index

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), pp. 367–368

Author index

https://doi.org/10.15559/16-VMSTA34AI

Pub. online: 4 Jan 2017 Type: Author Index

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), p. 365

Description of the symmetric convex random closed sets as zonotopes from their Feret diameters

Saïd Rahmani Jean-Charles Pinoli Johan Debayle

https://doi.org/10.15559/16-VMSTA70

Pub. online: 3 Jan 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), pp. 325–364

Abstract

In this paper, the 2-D random closed sets (RACS) are studied by means of the Feret diameter, also known as the caliper diameter. More specifically, it is shown that a 2-D symmetric convex RACS can be approximated as precisely as we want by some random zonotopes (polytopes formed by the Minkowski sum of line segments) in terms of the Hausdorff distance. Such an approximation is fully defined from the Feret diameter of the 2-D convex RACS. Particularly, the moments of the random vector representing the face lengths of the zonotope approximation are related to the moments of the Feret diameter random process of the RACS.

An estimate for an expectation of the simultaneous renewal for time-inhomogeneous Markov chains

Vitaliy Golomoziy

https://doi.org/10.15559/16-VMSTA68

Pub. online: 23 Dec 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), pp. 315–323

Abstract

In this paper, we consider two time-inhomogeneous Markov chains ${X_{t}^{(l)}}$, $l\in \{1,2\}$, with discrete time on a general state space. We assume the existence of some renewal set C and investigate the time of simultaneous renewal, that is, the first positive time when the chains hit the set C simultaneously. The initial distributions for both chains may be arbitrary. Under the condition of stochastic domination and nonlattice condition for both renewal processes, we derive an upper bound for the expectation of the simultaneous renewal time. Such a bound was calculated for two time-inhomogeneous birth–death Markov chains.

Approximation of solutions of SDEs driven by a fractional Brownian motion, under pathwise uniqueness

Oussama El Barrimi Youssef Ouknine

https://doi.org/10.15559/16-VMSTA69

Pub. online: 20 Dec 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), pp. 303–313

Abstract

Our aim in this paper is to establish some strong stability properties of a solution of a stochastic differential equation driven by a fractional Brownian motion for which the pathwise uniqueness holds. The results are obtained using Skorokhod’s selection theorem.

Goodness-of-fit test in a multivariate errors-in-variables model

A X = B

Alexander Kukush Yaroslav Tsaregorodtsev

https://doi.org/10.15559/16-VMSTA67

Pub. online: 20 Dec 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), pp. 287–302

Abstract

We consider a multivariable functional errors-in-variables model $AX\approx B$, where the data matrices A and B are observed with errors, and a matrix parameter X is to be estimated. A goodness-of-fit test is constructed based on the total least squares estimator. The proposed test is asymptotically chi-squared under null hypothesis. The power of the test under local alternatives is discussed.

Drift parameter estimation in stochastic differential equation with multiplicative stochastic volatility

Meriem Bel Hadj Khlifa Yuliya Mishura Kostiantyn Ralchenko Mounir Zili

https://doi.org/10.15559/16-VMSTA66

Pub. online: 13 Dec 2016 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 4 (2016), pp. 269–285

Abstract

We consider a stochastic differential equation of the form

\[ dX_{t}=\theta a(t,X_{t})\hspace{0.1667em}dt+\sigma _{1}(t,X_{t})\sigma _{2}(t,Y_{t})\hspace{0.1667em}dW_{t}\]

with multiplicative stochastic volatility, where Y is some adapted stochastic process. We prove existence–uniqueness results for weak and strong solutions of this equation under various conditions on the process Y and the coefficients a, $\sigma _{1}$, and $\sigma _{2}$. Also, we study the strong consistency of the maximum likelihood estimator for the unknown parameter θ. We suppose that Y is in turn a solution of some diffusion SDE. Several examples of the main equation and of the process Y are provided supplying the strong consistency.

Workshop “Fractality and Fractionality”

Yuliya Mishura Georgiy Shevchenko

https://doi.org/10.15559/16-VMSTA65

Pub. online: 15 Nov 2016 Type: Meeting Report

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 3, Issue 3 (2016), pp. 209–211

Abstract

This short note is devoted to the “Fractality and Fractionality” workshop, held on 17–20 May 2016 in Lorentz Center (Leiden, Netherlands).

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Online ISSN: 2351-6054
Print ISSN: 2351-6046

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