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A functional limit theorem for random processes with immigration in the case of heavy tails

Alexander Marynych Glib Verovkin

https://doi.org/10.15559/17-VMSTA76

Pub. online: 12 Apr 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 2 (2017), pp. 93–108

Abstract

Let $(X_{k},\xi _{k})_{k\in \mathbb{N}}$ be a sequence of independent copies of a pair $(X,\xi )$ where X is a random process with paths in the Skorokhod space $D[0,\infty )$ and ξ is a positive random variable. The random process with immigration $(Y(u))_{u\in \mathbb{R}}$ is defined as the a.s. finite sum $Y(u)=\sum _{k\ge 0}X_{k+1}(u-\xi _{1}-\cdots -\xi _{k})\mathbb{1}_{\{\xi _{1}+\cdots +\xi _{k}\le u\}}$. We obtain a functional limit theorem for the process $(Y(ut))_{u\ge 0}$, as $t\to \infty $, when the law of ξ belongs to the domain of attraction of an α-stable law with $\alpha \in (0,1)$, and the process X oscillates moderately around its mean $\mathbb{E}[X(t)]$. In this situation the process $(Y(ut))_{u\ge 0}$, when scaled appropriately, converges weakly in the Skorokhod space $D(0,\infty )$ to a fractionally integrated inverse stable subordinator.

Asymptotic behaviour of non-isotropic random walks with heavy tails

Mark Kelbert Enzo Orsingher

https://doi.org/10.15559/17-VMSTA75

Pub. online: 6 Apr 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 79–89

Abstract

A random flight on a plane with non-isotropic displacements at the moments of direction changes is considered. In the case of exponentially distributed flight lengths a Gaussian limit theorem is proved for the position of a particle in the scheme of series when jump lengths and non-isotropic displacements tend to zero. If the flight lengths have a folded Cauchy distribution the limiting distribution of the particle position is a convolution of the circular bivariate Cauchy distribution with a Gaussian law.

Randomly stopped maximum and maximum of sums with consistently varying distributions

Ieva Marija Andrulytė Martynas Manstavičius Jonas Šiaulys

https://doi.org/10.15559/17-VMSTA74

Pub. online: 6 Mar 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 65–78

Abstract

Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables, and η be a counting random variable independent of this sequence. In addition, let $S_{0}:=0$ and $S_{n}:=\xi _{1}+\xi _{2}+\cdots +\xi _{n}$ for $n\geqslant 1$. We consider conditions for random variables $\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution functions of the random maximum $\xi _{(\eta )}:=\max \{0,\xi _{1},\xi _{2},\dots ,\xi _{\eta }\}$ and of the random maximum of sums $S_{(\eta )}:=\max \{S_{0},S_{1},S_{2},\dots ,S_{\eta }\}$ belong to the class of consistently varying distributions. In our consideration the random variables $\{\xi _{1},\xi _{2},\dots \}$ are not necessarily identically distributed.

L^{p} (p \geq 2)

-solutions of generalized BSDEs with jumps and monotone generator in a general filtration

M’hamed Eddahbi Imade Fakhouri Youssef Ouknine

https://doi.org/10.15559/17-VMSTA73

Pub. online: 7 Feb 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 25–63

Abstract

In this paper, we study multidimensional generalized BSDEs that have a monotone generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. First, we prove the existence and uniqueness of ${\mathbb{L}}^{p}(p\ge 2)$-solutions in the case of a fixed terminal time under suitable p-integrability conditions on the data. Then, we extend these results to the case of a random terminal time. Furthermore, we provide a comparison result in dimension 1.

Generalized fractional Brownian motion

Mounir Zili

https://doi.org/10.15559/16-VMSTA71

Pub. online: 16 Jan 2017 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 4, Issue 1 (2017), pp. 15–24

Abstract

We introduce a new Gaussian process, a generalization of both fractional and subfractional Brownian motions, which could serve as a good model for a larger class of natural phenomena. We study its main stochastic properties and some increments characteristics. As an application, we deduce the properties of nonsemimartingality, Hölder continuity, nondifferentiablity, and existence of a local time.

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.

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