Lithuanian Mathematical Journal最新文献

We improve a recent universality theorem for the Riemann zeta-function in short intervals due to Antanas Laurinčikas with respect to the length of these intervals. Moreover, we prove that the shifts can even have exponential growth. This research was initiated by two questions proposed by Laurinčikas in a problem session of a recent workshop on universality.

In this paper, we answer the questions posed by Gundersen and Yang about the entire solutions of a class of nonlinear homogeneous binomial differential equations and obtain explicit forms of all the entire solutions of this type of differential equations. Moreover, we provide some examples to demonstrate that the equation solutions we obtained are accurate.

We obtain an asymptotic formula for the smoothly weighted first moment of quadratic Dirichlet L-functions at central values, with explicit main terms and an error term that is “square-root” of the main term.

Let λ₂, λ₃, λ₄, λ₅ be nonzero real numbers, not all negative. Let (mathfrak{V}) be a well-spaced sequence. Assume that λ₂/λ₃ is irrational and algebraic, and δ > 0. Let (Eleft(mathfrak{V},N,delta right)) be the number of (upsilon in mathfrak{V}) with (upsilon le N) such that the Diophantine inequality (left|{lambda }_{2}{p}_{2}^{2}+{lambda }_{3}{p}_{3}^{3}+{lambda }_{4}{p}_{4}^{4}+{lambda }_{5}{p}_{5}^{5}-upsilon right|<{upsilon }^{-delta }) has no solution in primes p₂, p₃, p₄, p₅. In this paper, we prove that for any (varepsilon >0,Eleft(mathfrak{V},N,delta right)ll {N}^{1-19/378+2delta +varepsilon },) which refines the previous result.

We show the existence of nodal solutions of the second-order nonlinear boundary value problem

where λ > 0 is a parameter, p : [0, 1]×ℝ² → ℝ and g : ℝ →ℝ are continuous functions, and g(0) = 0. For a nonnegative integer k, we say that a solution is nodal if it has only simple zeros in (0, 1) and has exactly k-1 such zeros. Under some suitable conditions, we obtain that there exists λ_∗ > 0 (or λ^∗ > 0) such that for fixed k ∈ {1, 2,…}, problem (P) has at least one nodal solution for λ ∈ (k²π²/g_∞, λ^∗) (or λ ∈ (λ^∗, k²π²/g_∞)), where g_∞ = lim_|s|→∞ g(s)/s. The proof of our main results relies on the bifurcation technique.

We show that the distribution class ℒ(γ) 𝒪𝒮 is not closed under infinitely divisible distribution roots for γ > 0, that is, we provide some infinitely divisible distributions belonging to the class, whereas the corresponding Lévy distributions do not. In fact, one part of these Lévy distributions belonging to the class 𝒪ℒℒ(γ) have different properties, and the other parts even do not belong to the class 𝒪ℒ. Therefore, combining with the existing related results, we give a completely negative conclusion for the subject and Embrechts–Goldie conjecture. Then we discuss some interesting issues related to the results of this paper.

We prove the existence and uniqueness of solutions to the differential equations of higher order ({x}^{left(lright)}left(sright)+gleft(s,xleft(sright)right)=0,sin left[c,dright],) satisfying three-point boundary conditions that contain a nonhomogeneous term (xleft(cright)=0,{x}{prime}left(cright)=0,{x}^{^{primeprime} }left(cright)=0,dots {x}^{left(l-2right)}left(cright)=0,{x}^{left(l-2right)}left(dright)-{beta x}^{left(l-2right)}left(eta right)=upgamma ,) where (lge mathrm{3,0}le c<eta <d,) the constants (beta ,upgamma ) are real numbers, and (g:left[c,dright]times {mathbb{R}}to {mathbb{R}}) is a continuous function. By using finer bounds on the integral of kernel, the Banach and Rus fixed point theorems on metric spaces are utilized to prove the existence and uniqueness of a solution to the problem.

Abstract

We consider the partial-sum process ({sum }_{k=1}^{left[ntright]}{X}_{k}^{left(nright)},) where (left{{X}_{k}^{left(nright)}={sum }_{j=0}^{infty }{alpha }_{j}^{left(nright)}{xi }_{k-j}left(bleft(nright)right), kin {mathbb{Z}}right},) n ≥ 1, is a series of linear processes with tapered filter ({alpha }_{j}^{left(nright)}={alpha }_{j} {1}_{left{0le jlelambdaleft(nright)right}}) and heavy-tailed tapered innovations ξ_j(b(n)), j ∈ Z. Both tapering parameters b(n) and ⋋ (n) grow to ∞ as n→∞. The limit behavior of the partial-sum process (in the sense of convergence of finite-dimensional distributions) depends on the growth of these two tapering parameters and dependence properties of a linear process with nontapered filter a_i, i ≥ 0, and nontapered innovations. We consider the cases where b(n) grows relatively slowly (soft tapering) and rapidly (hard tapering) and all three cases of growth of ⋋(n) (strong, weak, and moderate tapering).

Abstract

Integral tests are found for the convergence of two Spitzer-type series associated with a class of weighted averages introduced by Jajte [On the strong law of large numbers, Ann. Probab., 31(1):409–412, 2003]. Our main theorems are valid for a large family of dependent random variables that are not necessarily identically distributed. As a byproduct, we improve the Marcinkiewicz–Zygmund strong law of large numbers for asymptotically almost negatively associated sequences due to Chandra and Ghosal [Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables Acta Math. Hung., 71(4):327–336, 1996]. We also complement two limit theorems recently derived by Anh et al. [TheMarcinkiewicz–Zygmund-type strong law of large numbers with general normalizing sequences, J. Theor. Probab., 34(1):331–348, 2021] and Thành [On a new concept of stochastic domination and the laws of large numbers, Test, 32(1):74–106, 2023]. The obtained results are new even when the summands are independent.

In the paper, we establish the complete convergence for weighted sums of random variables satisfying generalized Rosenthal-type inequalities. Our results partially extend some known results and weaken their conditions. As statistical applications, we study the nonparametric regression model and obtain the complete consistency of the weighted regression estimator for the unknown regression functions.