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1.
Precise Large Deviations of Random Sums in Presence of Negative Dependence and Consistent Variation 总被引:5,自引:0,他引:5
The study of precise large deviations for random sums is an important topic in insurance and finance. In this paper, we extend
recent results of Tang (Electron J Probab 11(4):107–120, 2006) and Liu (Stat Probab Lett 79(9):1290–1298, 2009) to random sums in various situations. In particular, we establish a precise large deviation result for a nonstandard renewal
risk model in which innovations, modelled as real-valued random variables, are negatively dependent with common consistently-varying-tailed
distribution, and their inter-arrival times are also negatively dependent. 相似文献
2.
We investigate precise large deviations for heavy-tailed random sums. We prove a general asymptotic relation in the compound
renewal risk model for consistently varying-tailed distributions. This model was introduced in [Q. Tang, C. Su, T. Jiang,
and J.S. Zang, Large deviation for heavy-tailed random sums in compound renewal model, Stat. Probab. Lett., 52:91–100, 2001] as a more practical risk model. The proof is based on the inequality found in [D. Fuk and S.V. Nagaev,
Probability for sums of independent random variables, Theory Probab. Appl., 16:600–675, 1971]. 相似文献
3.
??In this paper, precise large deviations of nonnegative,
non-identical distributions and negatively associated random variables are investigated.
Under certain conditions, the lower bound of the precise large deviations for the
non-random sum is solved and the uniformly asymptotic results for the corresponding
random sum are obtained. At the same time, we deeply discussed the compound renewal
risk model, in which we found that the compound renewal risk model can be equivalent
to renewal risk model under certain conditions. The relative research results of
precise large deviations are applied to the more practical compound renewal risk model,
and the theoretical and practical values are verified. In addition, this paper also
shows that the impact of this dependency relationship between random variables to
precise large deviations of the final result is not significant. 相似文献
4.
In this paper, we establish strong laws for weighted sums of identically distributed negatively associated random variables.
Marcinkiewicz-Zygmund’s strong law of large numbers is extended to weighted sums of negatively associated random variables.
Furthermore, we investigate various limit properties of Cesàro’s and Riesz’s sums of negatively associated random variables.
Some of the results in the i.i.d. setting, such as those in Jajte (Ann. Probab. 31(1), 409–412, 2003), Bai and Cheng (Stat. Probab. Lett. 46, 105–112, 2000), Li et al. (J. Theor. Probab. 8, 49–76, 1995) and Gut (Probab. Theory Relat. Fields 97, 169–178, 1993) are also improved and extended to the negatively associated setting.
相似文献
5.
V. V. Petrov 《Journal of Mathematical Sciences》2006,133(3):1342-1344
Almost sure estimates of the growth of sums of nonnegative random variables are established. A generalization of author's
result on the growth of sums of the indicators of arbitrary events is obtained. Bibliography: 10 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 237–241. 相似文献
6.
A. N. Frolov 《Journal of Mathematical Sciences》2009,159(3):376-383
We derive logarithmic asymptotics for probabilities of large deviations for compound Cox processes. We show that under appropriate
conditions, these asymptotics are the same as those for sums of independent random variables and processes with independent
increments. When these conditions fail, the asymptotics of large deviations probabilities for compound Cox processes are quite
different. Bibliography: 5 titles.
Translated from Zapiski Nauehnykh, Seminarov POMI, Vol. 361, 2008, pp. 167–181. 相似文献
7.
Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications 总被引:1,自引:0,他引:1
In this paper, we establish some Rosenthal type inequalities for maximum partial sums of asymptotically almost negatively
associated random variables, which extend the corresponding results for negatively associated random variables. As applications
of these inequalities, by employing the notions of residual Cesàro α-integrability and strong residual Cesàro α-integrability,
we derive some results on L
p convergence where 1 < p < 2 and complete convergence. In addition, we estimate the rate of convergence in Marcinkiewicz-Zygmund strong law for partial
sums of identically distributed random variables. 相似文献
8.
We give a shorter proof of Kanter’s (J. Multivariate Anal. 6, 222–236, 1976) sharp Bessel function bound for concentrations of sums of independent symmetric random vectors. We provide
sharp upper bounds for the sum of modified Bessel functions I0(x) + I1(x), which might be of independent interest. Corollaries improve concentration or smoothness bounds for sums of independent
random variables due to Čekanavičius & Roos (Lith. Math. J. 46, 54–91, 2006); Roos (Bernoulli, 11, 533–557, 2005), Barbour & Xia (ESAIM Probab. Stat. 3, 131–150, 1999), and Le Cam (Asymptotic Methods in Statistical Decision Theory. Springer, Berlin Heidelberg New York, 1986).
相似文献
9.
Ibrahim A. Ahmad 《Annals of the Institute of Statistical Mathematics》1982,34(1):351-358
Summary For a sequence of independent and identically distributed random vectors, with finite moment of order less than or equal to
the second, the rate at which the deviation between the distribution functions of the vectors of partial sums and maximums
of partial sums is obtained both when the sample size is fixed and when it is random, satisfying certain regularity conditions.
When the second moments exist the rate is of ordern
−1/4 (in the fixed sample size case). Two applications are given, first, we compliment some recent work of Ahmad (1979,J. Multivariate Anal.,9, 214–222) on rates of convergence for the vector of maximum sums and second, we obtain rates of convergence of the concentration
functions of maximum sums for both the fixed and random sample size cases. 相似文献
10.
A contribution to large deviations for heavy-tailed random sums 总被引:22,自引:0,他引:22
In this paper we consider the large deviations for random sums
, whereX
n,n⩾1 are independent, identically distributed and non-negative random variables with a common heavy-tailed distribution function
F, andN(t), t⩾0 is a process of non-negative integer-valued random variables, independent ofX
n,n⩾1. Under the assumption that the tail of F is of Pareto’s type (regularly or extended regularly varying), we investigate what
reasonable condition can be given onN(t), t⩾0 under which precise large deviation for S( t) holds. In particular, the condition we obtain is satisfied for renewal counting
processes. 相似文献
11.
华志强 《纯粹数学与应用数学》2015,(4):360-366
从保险的实际出发,研究服从长尾分布族(L族)上的多元风险模型中随机变量序列的部分和的精确大偏差,其中假设随机变量序列是一列延拓负相依(END)的、同分布的随机变量序列,利用基于求L族的精确大偏差的方法得到了随机变量部分和的渐近下界. 相似文献
12.
In this article, we obtain the large deviations and moderate deviations for negatively dependent (ND) and non-identically distributed random variables defined on (-∞, +∞). The results show that for some non-identical random variables, precise large deviations and moderate deviations remain insensitive to negative dependence structure. 相似文献
13.
14.
A. N. Frolov 《Journal of Mathematical Sciences》2007,145(2):4931-4937
We derive logarithmic asymtotics for probabilities of small deviations for compound Cox processes. We show that under appropriate
conditions, these asymptotics are the same as those for sums of independent random variables and processes with independent
increments. When these conditions do not hold, the asymptotics of small deviations for compound Cox processes are quite different.
Bibliography: 9 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 163–175. 相似文献
15.
We investigate the precise large deviations of random sums of negatively dependent random variables with consistently varying tails. We find out the asymptotic behavior of precise large deviations of random sums is insensitive to the negative dependence. We also consider the generalized dependent compound renewal risk model with consistent variation, which including premium process and claim process, and obtain the asymptotic behavior of the tail probabilities of the claim surplus process. 相似文献
16.
We study an asymptotic property of weighted sequences of nonnegative functions which extends and unifies previous results concerned with precise asymptotics. As applications, we prove precise asymptotics for partial sums of independent identically distributed classical, noncommutative and free random variables.
相似文献17.
In this paper we show that Uspensky's expansion theorem for the Poisson approximation of the distribution of sums of independent Bernoulli random variables can be rewritten in terms of the Poisson convolution semigroup. This gives rise to exact evaluations and simple remainder term estimations for the deviations of the distributions in study with respect to various probability metrics, generalizing results of Shorgin (1977, Theory Probab. Appl., 22, 846–850). Finally, we compare the sharpness of Poisson versus normal approximations. 相似文献
18.
A. E. Mikusheva 《Mathematical Notes》2000,68(3):355-362
This paper extends results on complete convergence in the law of large numbers for subsequences to the case of negatively
associated nonidentically distributed random variables.
Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 411–420, September, 2000. 相似文献
19.
D. A. Yarotskii 《Mathematical Notes》1999,66(3):372-383
A nonhomogeneous random walk on the grid ℤ1 with transition probabilities that differ from those of a certain homogeneous random walk only at a finite number of points
is considered. Trajectories of such a walk are proved to converge to trajectories of a certain generalized diffusion process
on the line. This result is a generalization of the well-known invariance principle for the sums of independent random variables
and Brownian motion.
Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 459–472, September, 1999. 相似文献
20.
Kei Takeuchi Akimichi Takemura 《Annals of the Institute of Statistical Mathematics》1987,39(1):307-324
Summary Distribution of sum of vectors of 0–1 random variables is discussed generalizing the univariate results obtained in our previous
article Takeuchi and Takemura (1987,Ann. Inst. Statist. Math.,39, 85–102). As in our previous article no assumption is made on the independence of the 0–1 random variables. 相似文献