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In this paper, we obtain some results for the asymptotic behavior of the tail probability of a random sum Sτ = ∑τk = 1Xk, where the summands Xk, k = 1, 2, …, are conditionally dependent random variables with a common subexponential distribution F, and the random number τ is a non negative integer-valued random variable, independent of {Xk: k ? 1}.  相似文献   

3.
Let {Xn, n ? 1} be a sequence of asymptotically almost negatively associated (AANA, for short) random variables which is stochastically dominated by a random variable X, and {dni, 1 ? i ? n, n ? 1} be a sequence of real function, which is defined on a compact set E. Under some suitable conditions, we investigate some convergence properties for weighted sums of AANA random variables, especially the Lp convergence and the complete convergence. As an application, the Marcinkiewicz–Zygmund-type strong law of large numbers for AANA random variables is obtained.  相似文献   

4.
Abstract

Let {Xn, n ? 1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {bni, 1 ? i ? n, n ? 1} be an array of real numbers. In this article, we study the strong law of large numbers for the weighted sums ∑ni = 1bniXi without identical distribution. We present some sufficient conditions to prove the strong law of large numbers. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for NSD random variables is obtained. In addition, the complete convergence for the weighted sums of NSD random variables is established. Our results generalize and improve some corresponding ones for independent random variables and negatively associated random variables.  相似文献   

5.
Let X1:n ≤ X2:n ≤···≤ Xn:n denote the order statistics of a sample of n independent random variables X1, X2,…, Xn, all identically distributed as some X. It is shown that if X has a log-convex [log-concave] density function, then the general spacing vector (Xk1:n, Xk2:n ? Xk1:n,…, Xkr:n ? Xkr?1:n) is MTP2 [S-MRR2] whenever 1 ≤ k1 < k2 <···< kr ≤ n and 1 ≤ r ≤ n. Multivariate likelihood ratio ordering of such general spacing vectors corresponding to two random samples is also considered. These extend some of the results in the literature for usual spacing vectors.  相似文献   

6.
In this paper we consider a sequence of independent continuous symmetric random variables X1, X2, …, with heavy-tailed distributions. Then we focus on limiting behavior of randomly weighted averages Sn = R(n)1X1 + ??? + R(n)nXn, where the random weights R(n)1, …, Rn(n) which are independent of X1, X2, …, Xn, are the cuts of (0, 1) by the n ? 1 order statistics from a uniform distribution. Indeed we prove that cnSn converges in distribution to a symmetric α-stable random variable with cn = n1 ? 1/α1/α(α + 1).  相似文献   

7.
Let X = {X1, X2, …} be a sequence of independent but not necessarily identically distributed random variables, and let η be a counting random variable independent of X. Consider randomly stopped sum Sη = ∑ηk = 1Xk and random maximum S(η) ? max?{S0, …, Sη}. Assuming that each Xk belongs to the class of consistently varying distributions, on the basis of the well-known precise large deviation principles, we prove that the distributions of Sη and S(η) belong to the same class under some mild conditions. Our approach is new and the obtained results are further studies of Kizinevi?, Sprindys, and ?iaulys (2016) and Andrulyt?, Manstavi?ius, and ?iaulys (2017).  相似文献   

8.
In this article we obtain some novel results on pairwise quasi-asymptotically independent (pQAI) random variables. Concretely speaking, let X1, …, Xn be n real-valued pQAI random variables, and W1, …, Wn be another n non negative and arbitrarily dependent random variables, but independent of X1, …, Xn. Under some mild conditions, we prove that W1X1, …, WnXn are still pQAI as well. Our result is in a general setting whether the primary random variables X1, …, Xn are heavy-tailed or not. Finally, a special case of above result is applied to risk theory for investigating the finite-time ruin probability for a discrete-time risk model with a wide type of dependence structure.  相似文献   

9.
In this article, let {X1, …, Xn} be a sequence of negatively associated random variables and {ani, 1 ? i ? n, n ? 1} be a triangular array of constants. Several almost sure convergence theorems for the weighted sums ∑ni = 1aniXi are established.  相似文献   

10.
Let {X, Xn; n ≥ 1} be a sequence of real-valued iid random variables, 0 < r < 2 and p > 0. Let D = { A = (ank; 1 ≤ kn, n ≥ 1); ank, ? R and supn, k |an,k| < ∞}. Set Sn( A ) = ∑nk=1an, kXk for A ? D and n ≥ 1. This paper is devoted to determining conditions whereby E{supn ≥ 1, |Sn( A )|/n1/r}p < ∞ or E{supn ≥ 2 |Sn( A )|/2n log n)1/2}p < ∞ for every A ? D. This generalizes some earlier results, including those of Burkholder (1962), Choi and Sung (1987), Davis (1971), Gut (1979), Klass (1974), Siegmund (1969) and Teicher (1971).  相似文献   

11.
Let X1,…, Xn be mutually independent non-negative integer-valued random variables with probability mass functions fi(x) > 0 for z= 0,1,…. Let E denote the event that {X1X2≥…≥Xn}. This note shows that, conditional on the event E, Xi-Xi+ 1 and Xi+ 1 are independent for all t = 1,…, k if and only if Xi (i= 1,…, k) are geometric random variables, where 1 ≤kn-1. The k geometric distributions can have different parameters θi, i= 1,…, k.  相似文献   

12.
In this article, we study large deviations for non random difference ∑n1(t)j = 1X1j ? ∑n2(t)j = 1X2j and random difference ∑N1(t)j = 1X1j ? ∑N2(t)j = 1X2j, where {X1j, j ? 1} is a sequence of widely upper orthant dependent (WUOD) random variables with non identical distributions {F1j(x), j ? 1}, {X2j, j ? 1} is a sequence of independent identically distributed random variables, n1(t) and n2(t) are two positive integer-valued functions, and {Ni(t), t ? 0}2i = 1 with ENi(t) = λi(t) are two counting processes independent of {Xij, j ? 1}2i = 1. Under several assumptions, some results of precise large deviations for non random difference and random difference are derived, and some corresponding results are extended.  相似文献   

13.
ABSTRACT

Let X, X1, X2, … be a sequence of strictly stationary φ-mixing random variables with EX = μ > 0. In this paper, we show that a self-normalized version of almost sure central limit theorem (ASCLT) holds under the assumptions that the mixing coefficients satisfy ∑n = 1φ1/2(2n) < ∞ and the weight sequence {dk} satisfies a mild growth condition similar to Kolmogorov’s condition for the LIL. This shows that logarithmic averages, used traditionally in ASCLT for products of sums, can be replaced by other averages, leading to considerably sharper results.  相似文献   

14.
The supremum of random variables representing a sequence of rewards is of interest in establishing the existence of optimal stopping rules. Necessary and sufficient conditions are given for existence of moments of supn(Xn ? cn) and supn(Sn ? cn) where X1, X2, … are i.i.d. random variables, Sn = X1 + … + Xn, and cn = (nL(n))1/r, 0 < r < 2, L = 1, L = log, and L = log log. Following Cohn (1974), “rates of convergence” results are used in the proof.  相似文献   

15.
ABSTRACT

Least squares estimator of the stability parameter ? ? |α| + |β| for a spatial unilateral autoregressive process Xk, ? = αXk ? 1, ? + βXk, ? ? 1 + ?k, ? is investigated and asymptotic normality with a scaling factor n5/4 is shown in the unstable case ? = 1. The result is in contrast to the unit root case of the AR(p) model Xk = α1Xk ? 1 + ??? + αpXk ? p + ?k, where the limiting distribution of the least squares estimator of the unit root parameter ? ? α1 + ??? + αp is not normal.  相似文献   

16.
ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (XT1, X2T, …, XTk, ZT)T = (X11, …, X1n, …, Xk1, …, Xkn, Z1, …, Zn)T and derive the distribution of concomitant of multivariate order statistics arising from X1, X2, …, Xk. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.  相似文献   

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18.
In this paper a generalization of the semi-Pareto autoregressive minification process of the first order is given. The necessary and sufficient condition for stationarity of the process is determined. It is shown that the process is ergodic and uniformly mixing. The joint survival function and the joint density function of the random variables X n+h and X n are determined. The extremes of the random variables X 1, X 2, ..., X n and the geometric extremes of random variables X 1, X 2, ..., X N are derived and their asymptotic distributions are discussed. The estimation of the parameters is discussed and some numerical results are given.  相似文献   

19.
20.
Consider the randomly weighted sums Sm(θ) = ∑mi = 1θiXi, 1 ? m ? n, and their maxima Mn(θ) = max?1 ? m ? nSm(θ), where Xi, 1 ? i ? n, are real-valued and dependent according to a wide type of dependence structure, and θi, 1 ? i ? n, are non negative and arbitrarily dependent, but independent of Xi, 1 ? i ? n. Under some mild conditions on the right tails of the weights θi, 1 ? i ? n, we establish some asymptotic equivalence formulas for the tail probabilities of Sn(θ) and Mn(θ) in the case where Xi, 1 ? i ? n, are dominatedly varying, long-tailed and subexponential distributions, respectively.  相似文献   

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