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

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.

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.

We give new proofs of some known results on the values of the Riemann zeta function at positive integers and obtain some new theorems related to these values. Considering even zeta values as ζ(2n) = ηnπ2n, we obtain the generating functions of the sequences ηn and (−1)nηn. Using the Riemann–Lebesgue lemma, we give recurrence relations for ζ(2n) and ζ(2n + 1). Furthermore, we prove some series equations

We prove that, for given positive numbers α and h, the Riemann zeta-function ζ can approximate any nonvanishing analytic function on a simply connected compact subset of the right open half of the critical strip by shifts of type ζ(s + ih⌊αn⌋).

In this paper, we study the value distribution of L-functions in the Selberg class and concentrate on the uniqueness questions of L-functions that share one or two sets with an arbitrary meromorphic function. The results obtained in this paper improve the recent results due to Q.-Q. Yuan, X.-M. Li, and H.-X. Yi [Value distribution of L-functions and uniqueness questions of F. Gross, Lith. Math. J.

Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth normalized Fourier coefficient (Hecke eigenvalue) by λf (n). Let a(n) be the function with squarefull kernel. In this paper, we establish that \( {\sum}_{n\leqslant x}a(n){\lambda}_f^2\left(n+1\right)= Cx+O\left({x}^{13/14+\upepsilon}\right) \), where C is a constant that can be explicitly

In the paper, we consider the partial-sum process \( {\sum}_{k=1}^{\left[ nt\right]}{X}_k^{(n)}, \) where \( \left\{{X}_n^{(n)},k\in \mathbb{Z}\right\},n\ge 1, \) is a series of linear processes with innovations having heavy-tailed tapered distributions with tapering parameter bn depending on n. We show that, depending on the properties of a filter of a linear process under consideration and on the