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Asymptotic normality of local linear regression estimator for mixtures with varying concentrations

Daniel Horbunov

Rostyslav Maiboroda

https://doi.org/10.15559/25-VMSTA282

Pub. online: 2 Sep 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 13, Issue 1 (2026), pp. 1–17

Abstract

Finite mixtures with different regression models for different mixture components naturally arise in statistical analysis of biological and sociological data. In this paper a model of mixtures with varying concentrations is considered in which the mixing probabilities are different for different observations. The modified local linear regression estimator (mLLRE) is considered for nonparametric estimation of the unknown regression function for the given component of mixture. The asymptotic normality of the mLLRE is proved in the case when the regressor’s probability density function has jumps. Theoretically optimal bandwidth is derived. Simulations were made to estimate the accuracy of the normal approximation.

Laws of the iterated logarithm for iterated perturbed random walks

Oksana Braganets

https://doi.org/10.15559/25-VMSTA283

Pub. online: 28 Aug 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 13, Issue 1 (2026), pp. 19–38

Abstract

Let ${({\xi _{k}},{\eta _{k}})_{k\ge 1}}$ be independent identically distributed random vectors with arbitrarily dependent positive components and ${T_{k}}:={\xi _{1}}+\cdots +{\xi _{k-1}}+{\eta _{k}}$ for $k\in \mathbb{N}$. The random sequence ${({T_{k}})_{k\ge 1}}$ is called a (globally) perturbed random walk. Consider a general branching process generated by ${({T_{k}})_{k\ge 1}}$ and let ${Y_{j}}(t)$ denote the number of the jth generation individuals with birth times $\le t$. Assuming that $\mathrm{Var}\hspace{0.1667em}{\xi _{1}}\in (0,\infty )$ and allowing the distribution of ${\eta _{1}}$ to be arbitrary, a law of the iterated logarithm (LIL) is proved for ${Y_{j}}(t)$. In particular, an LIL for the counting process of ${({T_{k}})_{k\ge 1}}$ is obtained. The latter result was previously established in the article by Iksanov, Jedidi and Bouzeffour (2017) under the additional assumption that $\mathbb{E}{\eta _{1}^{a}}\lt \infty $ for some $a\gt 0$. In this paper, it is shown that the aforementioned additional assumption is not needed.

A change of measures technique for compound mixed renewal processes with applications in Risk Theory

Spyridon M. Tzaninis

Nikolaos D. Macheras

https://doi.org/10.15559/25-VMSTA281

Pub. online: 12 Aug 2025 Type: Research Article

Open Access

Journal: Modern Stochastics: Theory and Applications Volume 12, Issue 4 (2025), pp. 471–504

Abstract

Given a compound mixed renewal process S under a probability measure P, we provide a characterization of all progressively equivalent martingale probability measures Q on the domain of P, that convert S into a compound mixed Poisson process. This result extends earlier works of Delbaen and Haezendonck, Lyberopoulos and Macheras, and the authors, and enables us to find a wide class of price processes satisfying the condition of no free lunch with vanishing risk. Implications to the ruin problem and to the computation of premium calculation principles in an arbitrage-free insurance market are also discussed.

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 Volume 12, Issue 4 (2025), pp. 433–469

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].

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 Volume 12, Issue 4 (2025), pp. 407–431

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.

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 Volume 12, Issue 4 (2025), pp. 393–405

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.

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 Volume 12, Issue 4 (2025), pp. 375–392

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.

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.

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.

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