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Limit theorems for random Dirichlet series: boundary case

Alexander Iksanov

Ruslan Kostohryz

https://doi.org/10.15559/25-VMSTA276

Pub. online: 25 Mar 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

Buraczewski et al. (2023) proved a functional limit theorem (FLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series ${\textstyle\sum _{k\ge 2}}\frac{{(\log k)^{\alpha }}}{{k^{1/2+s}}}{\eta _{k}}$ as $s\to 0+$, where $\alpha \gt -1/2$ and ${\eta _{1}},{\eta _{2}},\dots $ are independent identically distributed random variables with zero mean and finite variance. A FLT and a LIL are proved in a boundary case $\alpha =-1/2$. The boundary case is more demanding technically than the case $\alpha \gt -1/2$. A FLT and a LIL for ${\textstyle\sum _{p}}\frac{{\eta _{p}}}{{p^{1/2+s}}}$ as $s\to 0+$, where the sum is taken over the prime numbers, are stated as the conjectures.

Exit times for some nonlinear autoregressive processes

Göran Högnäs Brita Jung

https://doi.org/10.15559/25-VMSTA277

Pub. online: 25 Mar 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

The expected exit time from the interval $[-1,1]$ is investigated for an autoregressive process defined recursively by

\[ {X_{n+1}^{\varepsilon }}=f\big({X_{n}^{\varepsilon }}\big)+\varepsilon {\xi _{n+1}},\hspace{1em}n=0,1,2,\dots ,\hspace{2.5pt}{X_{0}}=0.\]

Here, ε is a small positive parameter, $f:\mathbb{R}\mapsto \mathbb{R}$ is usually a contractive function and ${\{{\xi _{n}}\}_{n\ge 1}}$ is a sequence of i.i.d. random variables. In this paper, previous results for a linear function $f(x)=ax$ are extended to more general cases, with the main focus on piecewise linear functions.

Approximations related to tempered stable distributions

Kalyan Barman

Neelesh S Upadhye Palaniappan Vellaisamy

https://doi.org/10.15559/25-VMSTA275

Pub. online: 27 Feb 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

In this article, we first obtain, for the Kolmogorov distance, an error bound between a tempered stable and a compound Poisson distribution (CPD) and also an error bound between a tempered stable and an α-stable distribution via Stein’s method. For the smooth Wasserstein distance, an error bound between two tempered stable distributions (TSDs) is also derived. As examples, we discuss the approximation of a TSD to normal and variance-gamma distributions (VGDs). As corollaries, the corresponding limit theorem follows.

Investigation of sample paths properties of sub-Gaussian type random fields, with application to stochastic heat equations

Olha Hopkalo Lyudmyla Sakhno

https://doi.org/10.15559/25-VMSTA273

Pub. online: 27 Feb 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

The paper presents bounds for the distributions of suprema for a particular class of sub-Gaussian type random fields defined over spaces with anisotropic metrics. The results are applied to random fields related to stochastic heat equations with fractional noise: bounds for the tail distributions of suprema and estimates for the rate of growth are provided for such fields.

Strong laws of large numbers for lightly trimmed sums of generalized Oppenheim expansions

Rita Giuliano

Milto Hadjikyriakou

https://doi.org/10.15559/25-VMSTA272

Pub. online: 13 Feb 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

In the framework of generalized Oppenheim expansions, almost sure convergence results for lightly trimmed sums are proven. First, a particular class of expansions is identified for which a convergence result is proven assuming that only the largest summand is deleted from the sum; this result generalizes a strong law recently proven for the Lüroth digits and also covers some new cases that have never been studied before. Next, any assumptions concerning the structure of the Oppenheim expansions are dropped and a result concerning trimmed sums is proven when at least two summands are trimmed; combining this latter theorem with the asymptotic behavior of the r-th maximum term of the expansion, a convergence result is obtained for the case in which only the largest summand is deleted from the sum.

About stability of equilibria of one system of stochastic delay differential equations with exponential nonlinearity

Leonid Shaikhet

https://doi.org/10.15559/25-VMSTA274

Pub. online: 13 Feb 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

A system of two nonlinear delay differential equations under stochastic perturbations is considered. Nonlinearity of the exponential type in each equation of the system under consideration depends on the both variables of the system. The stability in probability of the zero and nonzero equilibria of the system is studied via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The obtained results are illustrated via examples and figures with numerical simulations of solutions of a considered system of stochastic differential equations. The proposed way of investigation can be applied to nonlinear systems of higher dimension and with other types of nonlinearity, both for delay differential equations and for difference equations.

Skorokhod

M_{1}

convergence of maxima of multivariate linear processes with heavy-tailed innovations and random coefficients

Danijel Krizmanić

https://doi.org/10.15559/25-VMSTA271

Pub. online: 28 Jan 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

In this paper, functional convergence is derived for the partial maxima stochastic processes of multivariate linear processes with weakly dependent heavy-tailed innovations and random coefficients. The convergence takes place in the space of ${\mathbb{R}^{d}}$-valued càdlàg functions on $[0,1]$ endowed with the weak Skorokhod ${M_{1}}$ topology.

Exponential utility maximization in small/large financial markets

Miklós Rásonyi

Hasanjan Sayit

https://doi.org/10.15559/24-VMSTA270

Pub. online: 23 Jan 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 2 (2025), pp. 225–250

Abstract

Obtaining a utility-maximizing optimal portfolio in a closed form is a challenging issue when the return vector follows a more general distribution than the normal one. In this paper, for markets based on finitely many assets, a closed-form expression is given for optimal portfolios that maximize an exponential utility function when the return vector follows normal mean-variance mixture models. Especially, the used approach expresses the closed-form solution in terms of the Laplace transformation of the mixing distribution of the normal mean-variance mixture model and no distributional assumptions on the mixing distribution are made.

Also considered are large financial markets based on normal mean-variance mixture models, and it is shown that the optimal exponential utilities in small markets converge to the optimal exponential utility in the large financial market. This shows, in particular, that to reach the best utility level investors need to diversify their investments to include infinitely many assets into their portfolio, and with portfolios based on only finitely many assets they will never be able to reach the optimum level of utility.

Noncentral moderate deviations for time-changed Lévy processes with inverse of stable subordinators

Antonella Iuliano

Claudio Macci

Alessandra Meoli

https://doi.org/10.15559/24-VMSTA269

Pub. online: 31 Dec 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 2 (2025), pp. 203–224

Abstract

This paper presents some extensions of recent noncentral moderate deviation results. In the first part, the results in [Statist. Probab. Lett. 185, Paper No. 109424, 8 pp. (2022)] are generalized by considering a general Lévy process $\{S(t):t\ge 0\}$ instead of a compound Poisson process. In the second part, it is assumed that $\{S(t):t\ge 0\}$ has bounded variation and is not a subordinator; thus $\{S(t):t\ge 0\}$ can be seen as the difference of two independent nonnull subordinators. In this way, the results in [Mod. Stoch. Theory Appl. 11, 43–61] for Skellam processes are generalized.

Regularity of paths of stochastic measures

Vadym Radchenko

https://doi.org/10.15559/24-VMSTA268

Pub. online: 26 Nov 2024 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 2 (2025), pp. 189–201

Abstract

Random functions $\mu (x)$, generated by values of stochastic measures are considered. The Besov regularity of the continuous paths of $\mu (x)$, $x\in {[0,1]^{d}}$, is proved. Fourier series expansion of $\mu (x)$, $x\in [0,2\pi ]$, is obtained. These results are proved under weaker conditions than similar results in previous papers.

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