Let {Xn}n >= 1 be a sequence of strictly stationary m-dependent random variables with zero means. In this article, we obtain an almost sure central limit theorem for partial sums of m-dependent random variables that are different from standardized partial sums, and by the way, we also establish a central limit theorem for the same. Under suitable conditions, we can also obtain consistent results for self-normalized partial sums.

Measuring and testing tail dependence is important in finance, insurance, and risk management. This paper proposes two tail dependence matrices based on classic rank correlation coefficients, which possess the desired population properties and interpretability. Their nonparametric estimators with strong consistency and asymptotic distributions are derived using the limit theory of U-processes. The simulation and application studies show that, compared to the tail dependence matrix based on Spearman’s ρ with large deviation, the Kendall-based tail dependence measure has stable variances under different tail conditions; thus, it is an effective approach to testing and quantifying tail dependence between random variables.

. Motivated by the questions posed by W. V. Li and A. Wei [Proc. Amer. Math. Soc. 137 (2009), pp. 195-204] and the conjecture of E. Lundberg and A. Thomack [On the average number of zeros of random harmonic polynomials with iid coefficients: precise asymptotics, Preprint, https://arxiv.org/ abs/2308.10333, 2023], we study the expected number of zeros of random harmonic polynomials Hn,m(z) = pn(z) + qm(z) with independently and identically distributed Gaussian coefficients. In this paper we verify the conjecture of E. Lundberg and A. Thomack that the expectation is O(n) when deg p = alpha deg q, where 0 <= alpha < 1. This result extends the previous estimates when m is a fixed constant or m = n to more general case.

In this article, we study the precise large deviations for sums of non identically distributed random variables {X-i,i >= 1} with distribution functions {F-i,i >= 1} that are widely upper orthant dependent (WUOD, in short) and phi-mixing. Moreover, the random variables are dependent, according to some copulas. We assume that the average of the right tails of distribution functions F-i is equivalent to the subexponential distribution F.

This paper considers a bidimensional delay-claim risk model with constant force of interest, in which each main claim may induce a delayed claim after a random time. Specifically, if the main claims and delayed claims follow the subexponential distributions with some dependence structure, we obtain some precise asymptotic estimates for the finite-time ruin probabilities. In addition, some numerical simulations are presented to test the performance of the theoretical results.

In this paper, we provide the coefficients of the Ramanujan expansion proposed by Wang [23] with a different method. Then, we present the rates of convergence of the expansion for the harmonic numbers, the values of the parameter h of the coefficients and some relevant inequalities. To demonstrate the superiority of our expansion with new coefficients over the classic Ramanujan formula, some numerical computations are also given.

In this article, we provide the recursive inequalities for weighted sums of Poisson random variables. Furthermore, we establish their upper and lower bounds. Some numerical simulations are also demonstrated. Particularly, in some cases, we demonstrate that our bounds are sharper than the ones in Pelekis (2016) by numerical results.

In this article, we study complete and complete integral convergence theorems for weighted sums of arrays of rowwise widely negative dependent random variables under the sub-linear expectations. The presented results are Baum-Katz type theorems that extend the corresponding results in classical probability space to the case of sub-linear expectation space.

Consider a by-claim risk model with a constant force of interest, where each main claim may induce a by-claim after a random time. We propose a time-claim-dependent framework, that incorporates dependence between not only the waiting time and the claim but also the main claim and the corresponding by-claim. Based on this framework, we derive some asymptotic estimates for the finite-time ruin probabilities in the case of subexponential claims. We also provide examples and verify the assumptions on dependence. Numerical studies are conducted to examine the performance of these asymptotic formulas.

In this paper,we obtain the general solution of Type Ⅰ McKay Distribution's Stein equation.We prove that the bounded solution of Type Ⅰ McKay Distribution's Stein equation ex-ists and it is unique.Furthermore,some properties of the unique bounded solution are provided,which could be useful for Type Ⅰ McKay Distribution's approximation.

In this paper we consider a special class of population-size-dependent branching processes, which can also be seen as an extension of Galton–Watson processes with state-dependent immigration (GWPSDI). The model is formulated as follows. Let Zn be the size of individuals belonging to the nth generation of a population. If Zn>1, the population evolves as a critical Galton–Watson process with finite variance; if Zn=1, the population evolves as another Galton–Watson process; if Zn=0, Zn+1 is drawn from a fixed immigration distribution. Based on the technical routes in Foster (1971) and Pakes (1971), some asymptotic results identical with those of GWPSDI are obtained by detailed computations.

In this paper, we study some asymptotic properties for the Bernstein estimators of the limit distribution function and the limit density function under a triangular sample. Specifically, we obtain the uniform strong consistency, mean squared error (MSE) and mean integrated squared error (MISE) for the resulting estimators. In addition, we give the optimal choice of the bandwidth parameter m in terms of the sample size n, for both the MSE and MISE. Numerical simulations are presented to show that the Bernstein estimators outperform Gaussian kernel estimators in terms of MISE under a triangular sample.

Hardy et al. (1934) came up with Hardy?s inequality in their book. Klaassen andWellner (2021) gave the probability version of the Hardy inequality when the parameter p > 1. Based on their work, in this paper, we assign the randomness to variables as well. When p > 1, we give some extensions of Hardy?s inequality. When 0 < p < 1, we provide the corresponding Hardy inequality in probability language. Also, we show that in some circumstances, our results contain the integral form of Hardy?s inequality. We give a reversed Hardy inequality for random variables as well.

Consider a multidimensional discrete-time risk model in terms of some dependence structure. For the case of multivariate regularly varying claims, we study the asymptotic behavior of the ruin probabilities determined by some ruin sets. The estimates hold uniformly for all time horizons. Then we focus on a ruin set that allows partial capital transfers. As an application, the optimal allocation of the initial reserve is obtained. Some numerical simulations are presented to illustrate the results.

The main purpose of this paper is to obtain the complete moment convergence for widely orthant dependent linear processes with random coefficients in form where is a sequence of stochastically dominated WOD random variables and is a sequence of random variables.

Consider a fractional Brownian motion starting at an interior point of the minimum and maximum parabolic domains, namely, D-min = {(x, y(1), y(2)) : parallel to x parallel to < min(j=1,2){(a(j) + y(j))(1/pj)}} and D-max = {(x, y(1), y(2)) parallel to x parallel to < max(j=1,2){(a(j)+y(j))(1/pj)}} in Rd+2, d >= 1, where parallel to center dot parallel to is the Euclidean norm in R-d, a(1), a(2) > 0, and p(1), p(2) > 1. Let tau(min) and tau(max) denote the first times that the fractional Brownian motion exits from D(min )and D-max, respectively. Asymptotically equivalent estimates of log P (tau(min) > t) and log P (tau(max) > t) are respectively given by using Gordon's inequality, depending on the relationship between p(1) and p(2). The proof methods are based on early works of Li, Shi, Lifshits, Aurzada and Lu. (C) 2022 Elsevier B.V. All rights reserved.

In this paper, we consider the Bernstein polynomial of the empirical distribution function F-n under a triangular sample, which we denote by (F) over cap (m,n). For the recentered and normalised statistic n(1/2) ((F) over cap (m,n)(x) - E-Gn(F) over cap (m,n)(x)), where x is defined on the interval (0, 1), the stochastic convergence to a Brownian bridge is derived. The main technicality in proving the normality is drawn off into a stochastic equicontinuity condition. To obtain the equicontinuity, we derive the uniform law of large numbers (ULLN) over a class of functions sup(H) |(P-n - E-Gn)h| by domination conditions of random covering numbers and covering integrals. In addition, we also derive the asymptotic covariance matrix for biavariant vector of Bernstein estimators. Finally, numerical simulations are presented to verify the validity of our main results.

In this paper, we consider the ruin probability of risk models with a geometric distribution of claim sizes. Since their probabilities can?t be calculated directly, we use exponential distribution to estimate its upper and lower bounds and asymptotic estimates based on the relationship between geometric distribution and exponential distribution. Finally, some numerical simulations are given to prove the superiority of our estimates.

Consider a renewal risk model in which the current inter-arrival time depends on a fixed number of previous claims but is independent of all other claims. We are interested in the precise large deviations of the aggregate amount of claims for the case of sub-exponential claims. In addition, an asymptotic estimate for the expectation of the aggregate amount of claims is also obtained. The asymptotic formulas are fully consistent with existing works in this study.

In this paper, the complete convergence and the complete moment convergence for maximal randomly weighted sums of widely orthant-dependent (WOD, in short) random variables are investigated. By using the main results, we further consider three striking and useful applications in probability and statistics. Our results generalize the corresponding earlier ones.