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Models of space-time random fields on the sphere

Mirko D’Ovidio Enzo Orsingher Lyudmyla Sakhno

https://doi.org/10.15559/22-VMSTA200

Pub. online: 3 Feb 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 2 (2022), pp. 139–156

Abstract

General models of random fields on the sphere associated with nonlocal equations in time and space are studied. The properties of the corresponding angular power spectrum are discussed and asymptotic results in terms of random time changes are found.

Averaging principle for the one-dimensional parabolic equation driven by stochastic measure

Boris Manikin

https://doi.org/10.15559/21-VMSTA195

Pub. online: 10 Jan 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 2 (2022), pp. 123–137

Abstract

A stochastic parabolic equation on $[0,T]\times \mathbb{R}$ driven by a general stochastic measure is considered. The averaging principle for the equation is established. The convergence rate is compared with other results on related topics.

Interacting Brownian motions in infinite dimensions related to the origin of the spectrum of random matrices

Yosuke Kawamoto

https://doi.org/10.15559/21-VMSTA193

Pub. online: 10 Jan 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 1 (2022), pp. 89–122

Abstract

The generalised sine random point field arises from the scaling limit at the origin of the eigenvalues of the generalised Gaussian ensembles. We solve an infinite-dimensional stochastic differential equation (ISDE) describing an infinite number of interacting Brownian particles which is reversible with respect to the generalised sine random point field. Moreover, finite particle approximation of the ISDE is shown, that is, a solution to the ISDE is approximated by solutions to finite-dimensional SDEs describing finite-particle systems related to the generalised Gaussian ensembles.

On path-dependent SDEs involving distributional drifts

Alberto Ohashi Francesco Russo

Alan Teixeira

https://doi.org/10.15559/21-VMSTA197

Pub. online: 3 Jan 2022 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 1 (2022), pp. 65–87

Abstract

The paper presents the study on the existence and uniqueness (strong and in law) of a class of non-Markovian SDEs whose drift contains the derivative in the sense of distributions of a continuous function.

Applications of a change of measures technique for compound mixed renewal processes to the ruin problem

Spyridon M. Tzaninis

https://doi.org/10.15559/21-VMSTA192

Pub. online: 23 Dec 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 1 (2022), pp. 45–64

Abstract

In the present paper the change of measures technique for compound mixed renewal processes, developed in Tzaninis and Macheras [ArXiv:2007.05289 (2020) 1–25], is applied to the ruin problem in order to obtain an explicit formula for the probability of ruin in a mixed renewal risk model and to find upper and lower bounds for it.

Asymptotic genealogies for a class of generalized Wright–Fisher models

Thierry Huillet Martin Möhle

https://doi.org/10.15559/21-VMSTA196

Pub. online: 15 Dec 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 1 (2022), pp. 17–43

Abstract

A class of Cannings models is studied, with population size N having a mixed multinomial offspring distribution with random success probabilities ${W_{1}},\dots ,{W_{N}}$ induced by independent and identically distributed positive random variables ${X_{1}},{X_{2}},\dots $ via ${W_{i}}:={X_{i}}/{S_{N}}$, $i\in \{1,\dots ,N\}$, where ${S_{N}}:={X_{1}}+\cdots +{X_{N}}$. The ancestral lineages are hence based on a sampling with replacement strategy from a random partition of the unit interval into N subintervals of lengths ${W_{1}},\dots ,{W_{N}}$. Convergence results for the genealogy of these Cannings models are provided under assumptions that the tail distribution of ${X_{1}}$ is regularly varying. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained by Huillet [J. Math. Biol. 68 (2014), 727–761] for the case when ${X_{1}}$ is Pareto distributed and complement those obtained by Schweinsberg [Stoch. Process. Appl. 106 (2003), 107–139] for models where sampling is performed without replacement from a supercritical branching process.

Covariance between the forward recurrence time and the number of renewals

Sotirios Losidis

https://doi.org/10.15559/21-VMSTA194

Pub. online: 15 Dec 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 9, Issue 1 (2022), pp. 1–16

Abstract

Recurrence times and the number of renewals in $(0,t]$ are fundamental quantities in renewal theory. Firstly, it is proved that the upper orthant order for the pair of the forward and backward recurrence times may result in NWUC (NBUC) interarrivals. It is also demonstrated that, under DFR interarrival times, the backward recurrence time is smaller than the forward recurrence time in the hazard rate order. Lastly, the sign of the covariance between the forward recurrence time and the number of renewals in $(0,t]$ at a fixed time point t and when $t\to \infty $ is studied assuming that the interarrival distribution belongs to certain ageing classes.

Principal components analysis for mixtures with varying concentrations

Olena Sugakova Rostyslav Maiboroda

https://doi.org/10.15559/21-VMSTA191

Pub. online: 12 Nov 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 4 (2021), pp. 509–523

Abstract

Principal Component Analysis (PCA) is a classical technique of dimension reduction for multivariate data. When the data are a mixture of subjects from different subpopulations one can be interested in PCA of some (or each) subpopulation separately. In this paper estimators are considered for PC directions and corresponding eigenvectors of subpopulations in the nonparametric model of mixture with varying concentrations. Consistency and asymptotic normality of obtained estimators are proved. These results allow one to construct confidence sets for the PC model parameters. Performance of such confidence intervals for the leading eigenvalues is investigated via simulations.

Modeling temporally uncorrelated components of complex-valued stationary processes

Niko Lietzén

Lauri Viitasaari Pauliina Ilmonen

https://doi.org/10.15559/21-VMSTA190

Pub. online: 10 Nov 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 4 (2021), pp. 475–508

Abstract

A complex-valued linear mixture model is considered for discrete weakly stationary processes. Latent components of interest are recovered, which underwent a linear mixing. Asymptotic properties are studied of a classical unmixing estimator which is based on simultaneous diagonalization of the covariance matrix and an autocovariance matrix with lag τ. The main contributions are asymptotic results that can be applied to a large class of processes. In related literature, the processes are typically assumed to have weak correlations. This class is extended, and the unmixing estimator is considered under stronger dependency structures. In particular, the asymptotic behavior of the unmixing estimator is estimated for both long- and short-range dependent complex-valued processes. Consequently, this theory covers unmixing estimators that converge slower than the usual $\sqrt{T}$ and unmixing estimators that produce non-Gaussian asymptotic distributions. The presented methodology is a powerful preprocessing tool and highly applicable in several fields of statistics.

Bounded in the mean solutions of a second-order difference equation

Mykhailo Horodnii Victoriia Kravets

https://doi.org/10.15559/21-VMSTA189

Pub. online: 9 Sep 2021 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 8, Issue 4 (2021), pp. 465–473

Abstract

Sufficient conditions are given for the existence of a unique bounded in the mean solution to a second-order difference equation with jumps of operator coefficients in a Banach space. The question of the proximity of this solution to the stationary solution of the corresponding difference equation with constant operator coefficients is studied.

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Online ISSN: 2351-6054
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