We study the asymptotic behavior of almost surely extreme values of discrete random variables. We give applications to birth and death processes and processes describing the length of the queue.

We develop a new robust empirical likelihood-based test for comparing the trimmed means of two independent populations. The simulation results indicate that the test has asymptotically correct level under various data distributions and controls the Type I error adequately for medium-size samples. For nonnormal data distributions, the power of the test is comparable to robust alternatives like Yuen’s

We consider the limit behavior of partial-sum random field \( {S}_{\mathrm{n}}\left({t}_1,{t}_2;X\left({b}_{\mathrm{n}}\right)\right)={\sum}_{k=1}^{\left[{n}_1{t}_1\right]}{\sum}_{l=1}^{\left[{n}_2{t}_2\right]}{X}_{k,l}^{\left(\mathrm{n}\right)} \), where\( \left\{{X}_{k,l}^{\left(\mathrm{n}\right)}={\sum}_{i=0}^{\infty }{\sum}_{j=0}^{\infty }{c}_i{,}_j{\upxi}_{k-i,l-j}\left({b}_{\mathrm{n}}\right)

We introduce the notions of λ-Baire property and λ-semiopen set using sets of Lebesgue measure zero. For a family A of subsets of the real line, we define the (λ∗)-property analogously as it was done in the category case for the (∗)-property. The main result is that the family A of all subsets of the real line having the λ-Baire property has the (λ∗)-property iff A is situated between the Euclidean

Given two integers a and k > 0, the number of factorizations of a (mod k) is the number of ordered pairs (s, t) ∈ {0, 1, . . . , k − 1}2 satisfying s · t ≡ a (mod k). This number is known to be expressible by a formula involving the greatest common divisor function. Motivated by such a formula, we derive several formulae counting the number of factorizations of a (mod k) subject to certain other natural

The purpose of the paper is to study the uniqueness problem of an L function in the Selberg class sharing one or two sets with an arbitrary meromorphic function having only finitely many poles. We manipulate the notion of weighted sharing of sets to improve a result of Q.Q. Yuan, X.M. Li, and H.X. Yi [Value distribution of L-functions and uniqueness questions of F. Gross, Lith. Math. J., 58(2):249–262

Erratum to “Existence and nonexistence of the solutions to the Cauchy problem of quasilinear parabolic equation with a gradient term” by Mingjun Zhou and Yan Leng, 61(1):123–142, January, 2021

We investigate the heteroscedastic regression model Yni = g(xni) + σniεni, i = 1, . . . , n, where \( {\sigma}_{ni}^2 \) = f(uni), (xni, uni) are known fixed design points, g and f are unknown functions, and the errors εni are assumed to form a stationary α-mixing random variables. Under some mild conditions, we obtain the asymptotic normality for wavelet estimators of f, prove their the asymptotic

We study the argument of L(s, f) associated with holomorphic cusp form in both weight aspect and t-aspect. We prove that for −1 ≤ σ ≤ 2 and t ≥ 4, argL(s, f) ≪ log kt, where s = σ + it. Assuming the generalized Riemann hypothesis (GRH), we have arg L(s, f) ≪ log kt/log log kt for σ ≥ 1/2 and t ≥ 4.

Hecke conjectured that an explicit set of theta series obtained from a quaternion algebra defined over ℚ ramified at a prime N is a basis of a space of holomorphicmodular forms of weight 2 for the Hecke congruence group Γ0(N). However, Eichler noticed that Hecke’s conjecture is not true in general. Hence it is natural to ask the dimension of the subspace of M2(Γ0(N)) spanned by the theta series, and

This paper deals with the existence and non-xistence of the solutions to the Cauchy problem of a class of quasilinear parabolic equations with a gradient term. We establish Fujita-type blowup theorems and determine the critical Fujita exponent in terms of spatial dimension, the asymptotic behavior of the coefficients of the gradient term at infinity, the exponents of spatial positions in the coefficients

In this paper, we demonstrate a character analogue of the Lerch formula and multiplication formulas for the Dirichlet L-functions. These formulas produce multiplication formulas for the character gamma functions and related polynomials. Moreover, we establish closed-form evaluation formulas for the power series with Dirichlet L-value coefficients.

We say that an ideal ℐ has property (T) if for every ℐ-convergent series \( {\sum}_{n=1}^{\infty }{x}_n, \) there exists a set A ∈ ℐ such that Σn∈ℕ\A xn converges in the usual sense. The aim of this paper is to construct a nontrivial ideal with property (T) under the assumption that cov (ℳ) = c.

In this paper, by using linear forms in logarithms and the Baker–Davenport reduction procedure we prove that there are no even perfect numbers appearing in generalized Pell sequences. We also deduce some interesting results involving generalized Pell numbers, which we believe are of independent interest. This paper continues a previous work that searched for perfect numbers in the classical Pell sequence

This paper concerns a generalization of approximately continuous functions, namely \( \mathcal{S} \)-approximately continuous functions. This notion is associated with \( \mathcal{S} \)-density points, where \( \mathcal{S} \) is a sequence of measurable sets tending to 0. Moreover, we present some properties of these functions and show their connection with measurable functions, functions from the

In this paper, we apply the circle method to study the average behavior of the triple divisor function over values of quadratic form \( {n}_1^2+{n}_2^2+\cdots +{n}_l^2 \) with l ⩾ 3. We improve and generalize previous results.

The main purpose of this paper is studying some properties on the growth and value distribution of solutions of some difference equations. We obtain some results on the existence and estimates of growth of solutions f for some difference equations and some estimates of the exponent of convergence of poles of Δf, Δ2f, Δf/f, and Δ2f/f, which improve and extend the previous results given by Chen, Li,

We extend the results of Xh.Z. Krasniqi [Acta Comment. Univ. Tartu. Math., 17:89–101, 2013] and the authors [Acta Comment. Univ. Tartu. Math., 13:11–24, 2019] to the case where in the measures of estimations r-differences of the entries are used.

We establish the asymptotic degree distribution of the typical vertex of inhomogeneous and passive random intersection graphs under minimal moment conditions.

We introduce a new option pricing equation with noise in a frictional financial market, which is fully different from the classical option pricing equation, and arrive at the existence of martingale solutions of this option pricing equation regardless of incompressibility. Furthermore, we also discuss the uniqueness of martingale solutions.

We study the distribution of the zeros of the kth derivatives of the functions belonging to the extended Selberg class. We obtain the zero-free regions for these derivatives and a Riemann–von Mangoldt-type estimate of the count of their nontrivial zeros.

Given a sequence of independent standard Gaussian variables (Zn), the classical Pisier algebra P is the class of all continuous functions f on the unit circle T such that for each t ∈ 𝕋, the random Fourier series \( {\sum}_{n\in \mathrm{\mathbb{Z}}}{Z}_n\hat{f}(n)\times \exp \left(2\pi \mathrm{i} nt\right) \) converges in L2 and the corresponding sums constitute a Gaussian process that admits a continuous

Abstract. This research paper stands for the estimation of the precision matrix of the normal matrix with monotone missing data. We explicitly provide maximum and expectation a posteriori estimators. For this purpose, we basically use an extension of the Wishart distribution, that is, the Riesz distribution on symmetric matrices. We prove that some of the latter distributions may be presented using

Abstract. Let X = (X(t))t≥0 (X(0) = 0) be a continuous centered Gaussian process on a probability space (Ω,F,P), and let (Yt)t∈[0,1] (Y0 = 0) be a continuous process (on the same probability space) with nondecreasing paths, independent of X. Define the time-changed Gaussian process Zt = X(Yt), t ∈ [0, 1]. In this paper, we investigate a problem of finite-dimensional large deviations and a problem of

Abstract. We study an iterated temporal and contemporaneous aggregation of N independent copies of a strongly stationary subcritical Galton–Watson branching process with regularly varying immigration having index α ∈ (0, 2). We show that limits of finite-dimensional distributions of appropriately centered and scaled aggregated partial-sum processes exist when first taking the limit as N → ∞and then

Abstract. We show that, under the long-tailedness of the densities of normalized Lévy measures, the densities of infinitely divisible distributions on the half-line are subexponential if and only if the densities of their normalized Lévy measures are subexponential. Moreover, we prove that, under a certain continuity assumption, the densities of infinitely divisible distributions on the half-line are

For sequences of d + 1 signs + and − beginning with a + and having exactly two variations of sign, we give some sufficient conditions for the (non)existence of degree d real univariate polynomials with such signs of the coefficients and having given numbers of positive and negative roots compatible with Descartes’ rule of signs.

In this paper, we present a switching property for squared Bessel process. We prove that the starting point and the time parameter can be in a sense switched. As a consequence, we obtain new distributional dependencies among a squared Bessel process R, a Brownian motion with drift \( {B}^{\left(\mu \right)},\mu \ge 0,\left({B}_t^{\left(\mu \right)}:= {B}_t+\mu t\kern0.5em \mathrm{for}\ \mathrm{a}\

We consider themain boundary value problems of linear elastostatics with nonregular data. We prove existence and uniqueness results for bounded and exterior domains of ℝ3 of class Ck (k ⩾ 2). In the case of isotropic body, we prove the results for domains of class C1,α (α ∈ (0, 1]) and of class C1 in the case of the displacement problem.

We evaluate a weighted sum of Gauss hypergeometric functions for certain ranges of the argument, weights and parameters. We establish the domain of absolute convergence of this series by determining the growth of the hypergeometric function for large summation index. We present an application to Galton–Watson branching processes arising in the theory of stochastic processes. We introduce a new class

In the metadata of the article on SpringerLink, the corresponding author is incorrect. The corresponding author is Tran Loc Hung (tlhung@ufm.edu.vn; tlhungvn@gmail.com)

In this paper, we prove an existence and uniqueness theorem and a comparison theorem for a class of anticipated mean-field backward stochastic differential equations with jumps.

Let f be the characteristic function of a probability measure μf on ℝn, and let σ > 0. We study the following extrapolation problem: under what conditions on the neighborhood of infinity Vσ = {x ∈ ℝn : |xk| > σ, k = 1, …n} in ℝndoes there exist a characteristic function g on ℝn, such that g = f on Vσ but g ≢ f? Let μf have a nonzero absolutely continuous part with continuous density 𝜑. In this paper

We present upper bounds of the integral \( {\int}_{-\infty}^{\infty }{\left|x\right|}^l\left|\mathrm{P}\left\{{Z}_N0\left({S}_N{X}_1+\dots +{X}_N\right) \) of centered random variables X1,X2, . . . satisfying the uniformly strong mixing condition. The number of summands N is a nonnegative integer-valued random variable independent of X1,X2, . . . .

Let X and Y be two independent random variables with corresponding distributions F and G on [0,∞). The distribution of the product XY , which is called the product convolution of F and G, is denoted by H. In this paper, we give some suitable conditions on F and G, under which the distribution H belongs to the long-tailed distribution class. Here F is a generalized long-tailed distribution, not necessarily

We derive exponential bounds for the tail of the distribution of normalized sums of triangular arrays of random variables, not necessarily independent, under the law of ordinary logarithm. Furthermore, we provide estimates for partial sums of triangular arrays of independent random variables belonging to suitable grand Lebesgue spaces and having heavy-tailed distributions.

We consider nonparametric estimation of the ridge of a probability density function for multivariate linear processes with long-range dependence. We derive functional limit theorems for estimated eigenvectors and eigenvalues of the Hessian matrix. We use these results to obtain the weak convergence for the estimated ridge and asymptotic simultaneous confidence regions.

Investigating the same problems for linear random fields with tapered innovations, I realized that the Hurst index of the limit FBM process and conditions on tapering parameter γ in Theorem 1 were incorrectly calculated.

In this paper, we consider a strictly stationary sequence of m-dependent random variables through a compatible sequence of independent and identically distributed random variables by the moving averages processes. Using the Zolotarev distance, we estimate some rates of convergence in the weak limit theorems for normalized geometric random sums of the strictly stationary sequence of m-dependent random

We establish a central limit theorem for counting large continued fraction digits (an), that is, we count occurrences {an>bn}, where (bn) is a sequence of positive integers. Our result improves a similar result by Philipp, which additionally assumes that bn tends to infinity. Moreover, we give a refinement of the famous Borel–Bernstein theorem for continued fractions regarding the event that the nth

In this paper, we study the multiplicity of solutions for a class of nonhomogeneous Schrödinger–Poisson systems under general superlinear conditions at infinity. With the aid of Ekeland’s variational principle, Jeanjean’s monotone method, Pohožaev’s identity, and the mountain pass theorem, we prove that a Schrödinger–Poisson system has at least two positive solutions, which generalizes and improves

In this paper, we study some asymptotic properties of CFG estimator of the Pickands dependence function of strictly stationary absolutely regular sequences of bivariate extremes. We then propose an asymptotic test of independence of the vector margins. Finite sample properties of the estimate are investigated by simulation.

The aim of this paper is to present newupper bounds for the distance between a properly normalized permanent of a rectangular complex matrix and the product of the arithmetic means of the entries of its columns. It turns out that the bounds improve those from earlier work. Our proofs are based on some new identities for the above-mentioned difference and also for related expressions for matrices over

Consider a nonstandard renewal risk model in which claims and interarrival times form a sequence of independent and identically distributed random pairs, with each pair obeying arbitrary dependence or size-dependence structure. In the case of heavy-tailed claims, we obtain the asymptotic behavior of finite-time ruin probability with the uniformity in time in some infinite regions.

In this paper, we extend to CKLS and CEV processes a known result on weak approximation of CIR processes by discrete random variables. Namely, for CKLS and CEV processes, we construct first-order split-step weak approximations that use generation of two-valued random variables at each discretization step. The accuracy of constructed approximations is illustrated by several simulation examples.

In this paper, we consider the Dirichlet problem for a nonlinear wave equation with Balakrishnan–Taylor term. We establish the local existence is established by the linearization method together with the Faedo–Galerkin method. Next, we prove an exponential energy decay result by suitable Lyapunov functionals.

In this paper, we propose and analyze numerical treatment for a singularly perturbed convection–diffusion boundary value problem with nonlocal condition. First, the boundary layer behavior of the exact solution and its first derivative have been estimated. Then we construct a finite difference scheme on a uniform mesh. We prove the uniform convergence of the proposed difference scheme and give an error

Consider a continuous-time process {ZNt}, where {Zn} is a Galton–Watson process with offspring mean m, and {Nt} is a Poisson process independent of {Zn}. It turns out that Rt := ZNt+1/ZNt is an estimator of m. We deal with large deviation rates for the convergence of Rt to m for the supercritical and critical cases.

We explore asymptotically optimal bounds for deviations of distributions of independent Bernoulli random variables from the Poisson limit in terms of the Shannon relative entropy and Rényi/relative Tsallis distances (including Pearson’s _2). This part generalizes the results obtained in Part I and removes any constraints on the parameters of the Bernoulli distributions.

We propose an approach for forecasting risk contained in future observations in a time series. We take into account both the shape parameter and the extremal index of the data. This significantly improves the quality of risk forecasting over methods that are designed for i.i.d. observations and over the return level approach. We prove functional joint asymptotic normality of the common estimators of

In the paper, we investigate the intersection local time for two correlated Brownian motions on the plane that form a diffusion process in ℝ4 associated with a divergence-form generator. Using Gaussian heat kernel bounds, we prove the existence of intersection local time for these Brownian motions, obtain estimates of its moments, and establish the law of iterated logarithm for it.

Convexity plays a prominent role in a number of areas, but practical considerations often lead to nonconvex functions. We suggest a method for determining regions of convexity and also for assessing the lack of convexity of functions in the other regions. The method relies on a specially constructed decomposition of symmetric matrices. Illustrative examples accompany theoretical results.

This year professor Vygantas Paulauskas is celebrating his 75 birthday. An outstanding probabilist he is well known for his research in the field of probability limit theorems and mathematical statistics. He was the moving spirit behind the organization of the Vilnius conferences in Probability and Statistics 2008–2018. The conversation gives a glimpse into the mathematical life of our country in the

For a stationary sequence that is regularly varying and associated, we give conditions guaranteeing that partial sums of this sequence, under normalization related to the exponent of regular variation, converge in distribution to a stable non-Gaussian limit. The obtained limit theorem admits a natural extension to the functional convergence in Skorokhod’s M1 topology.

Let X 1 and N ≥ 0 be integer-valued power-law random variables. For a randomly stopped sum S N = X 1+⋯+ X N of independent and identically distributed copies of X 1, we establish a first-order asymptotics of the local probabilities P ( S N = t ) as t → +1 ∞. Using this result, we show the scaling k -δ, 0 ≤ δ ≤ 1, of the local clustering coefficient (of a randomly selected vertex of degree k ) in a

Let X , X 1, X 2, . . . be a sequence of nondegenerate i.i.d. random variables, let μ  = { μ ni  :  n  ∈  ℕ +,  i  = 1, …,  n } be a triangular array of possibly random probabilities on the interval [0, 1], and let \( \mathcal{F} \) be a class of functions with bounded q -variation on [0, 1] for some q ∈ [1, 2). We prove the asymptotic normality uniformly over \( \mathcal{F} \) of self-normalized weighted

We review recent results on anisotropic scaling limits and the scaling transition for linear and their subordinated nonlinear long-range dependent stationary random fields X on ℤ 2 . The scaling limits \( {V}_{\upgamma}^X \) are taken over rectangles in ℤ2 whose sides increase as O ( λ ) and O ( λγ ) as λ→∞ for any fixed γ > 0. The scaling transition occurs at \( {\upgamma}_0^X>0 \) provided that \(

We study the absolute continuity and local limit theorems for homogeneous functionals defined on configurations of point processes (p.p.s). For empirical p.p.s, we show that under mild hypotheses the distribution of such a functional has a density. Moreover, we present results on convergence in total variation of this distribution to some limit.

In this paper, we propose a new method of upper bounds for the number of integer polynomials of the fourth degree with a given discriminant. By direct calculation similar results were established by H. Davenport and D. Kaliada for polynomials of second and third degrees.

It is known that there exist only finitely many trivial solutions of a certain Diophantine equation over number fields except for the case the rational number field. In this paper, we calculate an explicit upper bound of the solution-free region.