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On parameter estimation for

N (μ, σ^{2} I_{3})

based on projected data into

S^{2}

Jordi-Lluís Figueras

Aron Persson

Lauri Viitasaari

https://doi.org/10.15559/25-VMSTA279

Pub. online: 17 Jun 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

The projected normal distribution, with isotropic variance, on the 2-sphere is considered using intrinsic statistics. It is shown that in this case, the expectation commutes with the projection, and that the covariance of the normal variable has a 1-1 correspondence with the intrinsic covariance of the projected normal distribution. This allows us to estimate, after the model identification, the parameters of the underlying normal distribution that generates the data.

On the reducibility of affine models with dependent Lévy factors

Michał Barski

Rafał Łochowski

https://doi.org/10.15559/25-VMSTA280

Pub. online: 17 Jun 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

The paper is devoted to the study of the short rate equation of the form

\[ \text{d}R(t)=F(R(t))\text{d}t+{\sum \limits_{i=1}^{d}}{G_{i}}(R(t-))\text{d}{Z_{i}}(t),\hspace{1em}R(0)={R_{0}}\ge 0,\hspace{3.33333pt}t\gt 0,\]

with deterministic functions $F,{G_{1}},\dots ,{G_{d}}$ and a multivariate Lévy process $Z=({Z_{1}},\dots ,{Z_{d}})$ with possibly dependent coordinates. This equation is assumed to have a nonnegative solution which generates an affine term structure model. Under some mild assumptions on the Lévy measure of Z it is shown that the same term structure is generated by an equation with affine drift term and noise being a one-dimensional α-stable process with index of stability $\alpha \in (1,2)$. For this case the shape of possible simple forward curves is characterized. A precise description of normal, inverse and humped profiles in terms of the equation coefficients and the stability index α is provided.

The paper generalizes the classical results on the Cox–Ingersoll–Ross (CIR) model [Econometrica 53 (1985), 385–408], as well as on its extended version where Z is a one-dimensional Lévy process [SIAM J. Financ. Math. 11(1) (2020), 131–147, Bond Markets with Lévy Factors, Cambridge University Press, 2020]. It is the starting point for the classification of affine models with dependent Lévy processes, in the spirit of [J. Finance 5 (2000), 1943–1978] and [Classification and calibration of affine models driven by independent Lévy processes, https://arxiv.org/abs/2303.08477].

Moments of Student’s t-distribution: a unified approach

Justin Lars Kirkby Dang H. Nguyen Duy Nguyen

https://doi.org/10.15559/25-VMSTA278

Pub. online: 25 Apr 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications

Abstract

In this paper, new closed form formulae for moments of the (generalized) Student’s t-distribution are derived in the one dimensional case as well as in higher dimensions through a unified probability framework. Interestingly, the closed form expressions for the moments of the Student’s t-distribution can be written in terms of the familiar Gamma function, Kummer’s confluent hypergeometric function, and the hypergeometric function. This work aims to provide a concise and unified treatment of the moments for this important distribution.

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 Volume 12, Issue 3 (2025), pp. 347–373

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 Volume 12, Issue 3 (2025), pp. 325–346

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 Volume 12, Issue 3 (2025), pp. 289–311

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.

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 Volume 12, Issue 3 (2025), pp. 313–324

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.

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 Volume 12, Issue 3 (2025), pp. 273–288

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.

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 Volume 12, Issue 3 (2025), pp. 251–272

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.

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