唐風(fēng)琴
(1. 蘭州大學(xué)數(shù)學(xué)與統(tǒng)計學(xué)院,甘肅 蘭州 730000;2. 淮北師范大學(xué)數(shù)學(xué)科學(xué)學(xué)院,安徽 淮北 235000)
非負(fù)相依隨機(jī)變量和的尾部概率一致漸近估計*
唐風(fēng)琴1,2
(1. 蘭州大學(xué)數(shù)學(xué)與統(tǒng)計學(xué)院,甘肅 蘭州 730000;2. 淮北師范大學(xué)數(shù)學(xué)科學(xué)學(xué)院,安徽 淮北 235000)
假設(shè){Xi}i≥1為一列非負(fù)不同分布的隨機(jī)變量,其分布函數(shù)屬于重尾子族-C族且聯(lián)合分布滿足多元FGMcopula函數(shù)。探討了序列{Xi}i≥1的部分和及隨機(jī)和的一致漸近估計,推廣了相依結(jié)構(gòu)隨機(jī)變量尾部漸近概率的相應(yīng)結(jié)果.。
精細(xì)大偏差;FGM;C族
(1)
下面介紹著名的Farlie-Gumbel-Morgenstern(FGM)copula函數(shù)。n維FGM分布具有如下形式:對所有實(shí)數(shù)x1,x2,…,xn有
(2)
其中Fi(x)是Xi的分布函數(shù),ajk為實(shí)數(shù)。若對任意的j,k,ajk=0,則序列x1,x2,…,xn是獨(dú)立的,本文假設(shè)至少存在一個ajk≠0。由Tang等[14]知若Fi(x)是連續(xù)的,則有
(3)
進(jìn)一步地,定義序列x1,x2,…,xn的生存函數(shù)為
(4)
β∈[-1,1]。注意到β=0時,Xi和Xj是獨(dú)立的。同時可得
(5)
滿足這種關(guān)系的序列又稱為尾上漸近獨(dú)立的(uppertailasymptoticindependent)。 因此,F(xiàn)GM包含了很多的相依結(jié)構(gòu),如正(負(fù))相依等?,F(xiàn)在給出本文的主要結(jié)果。
定理2 設(shè)隨機(jī)變量序列{Xi}i≥1滿足定理1中條件且對于任意的i,j,存在0 (6) 假設(shè)隨機(jī)過程{N(t),t≥0}與{Xi}i≥1獨(dú)立,且對任意小的δ>0,{N(t),t≥0}滿足 (7) 則對任意的γ>0,當(dāng)x≥γλ(t),t→∞時一致成立 首先給出在證明部分和大偏差的下界時一個非常重要的結(jié)論。 (8) 令 引理3得證。 其中第二個不等式由(4)式得到。 上界估計。對任意的0<υ<1,沿用引理3中的記號,有 (9) 同引理3的證明類似,令 (10) 其中C為常數(shù),倒數(shù)第三步用到引理2的結(jié)論。聯(lián)立(9)和(10)式,由υ的任意性及引理1可得當(dāng)x≥γn,n→∞時,上界估計得證。 I1+I2+I3 (11) 由δ的任意性,(6)式及定理1可得,當(dāng)x≥γλ(t),t→∞時有 (12) 其中C1=C1(c)為正常數(shù)。 與I1處理類似,當(dāng)x≥γλ(t),t→∞時, (13) 同理可得 (14) 將(12)-(14)式代入(11)式,定理2得證。 [1] 陳傳勇. 關(guān)于任意同分布隨機(jī)變量序列最大值不等式及其應(yīng)用[J]. 中山大學(xué)學(xué)報(自然科學(xué)版),2015, 52(2): 59-61. [2]MIKOSCHT,NAGAEVAV.Largedeviationsofheavy-tailedsumswithapplicationsininsurance[J].Extremes, 1998, 1(1): 81-110. [3]TANGQ,SUQ,JIANGT,etal.Largedeviationsforheavy-tailedrandomsumsincompoundrenewalmodel[J].StatisticsandProbabilityLetters, 2001, 52(1): 91-100. [4]LIUY,HUYJ.Largedeviationsforheavy-tailedrandomsumsofindependentrandomvariableswithdominatedlyvaryingtails[J].ScienceinChinaSeriesA, 2003, 46(3): 383-395. [5]NGKW,TANGQ,YANJA,etal.Preciselargedeviationsforsumsofrandomvariableswithconsistentlyvaryingtails[J].JournalofAppliedProbability, 2004(41): 93-107. [6]TANGQ.Insensitivitytonegativedependenceoftheasymptoticbehaviorofpreciselargedeviations[J].ElectronicJournalofProbability, 2006, 11: 107-120. [7]LIUL.Preciselargedeviationsfordependentrandomvariableswithheavytails[J].StatisticsandProbabilityLetters, 2009, 79(9): 1290-1298. [8]CHENY,YUENKC,NGKW.Preciselargedeviationsofrandomsumsinpresenceofnegativedependenceandconsistentvariation[J].MethodologyandComputinginAppliedProbability, 2011, 13(4):821-833. [9]WANGYB,WANGKY,CHENGDY.Preciselargedeviationsforsumsofnegativelyassociatedrandomvariableswithcommondominatedlyvaryingtails[J].ActaMathematicaSinica-EnglishSeries, 2006, 22(6): 725-1734. [10]WANGKY,YANGY,LINJG.Preciselargedeviationsforwidelyorthantdependentrandomvariableswithdominatedlyvaryingtails[J].FrontiersofMathematicsinChina, 2012, 7(5): 919-932. [11]TANGQ,TSITSIASHVILIG.Preciseestimatesfortheruinprobabilityinfinitehorizoninadiscretetimemodelwithheavy-tailedinsuranceandfinancialrisks[J].StochasticProcessesandTheirApplications, 2003, 108(2): 299-325. [12]TANGFQ,BAIJM.Preciselargedeviationsforaggregatelossprocessinamulti-riskmodel[J].JournaloftheKoreanMathematicalSociety, 2015, 52(3): 447-467. [13]HEW,CHENGD,WANGY.Asymptoticlowerboundsofpreciselargedeviationswithnonnegativeanddependentrandomvariables[J].StatisticsandProbabilityLetters, 2013, 83(1): 331-338. [14]TANGQ,VERNICR.Theimpactonruinprobabilitiesoftheassociationstructureamongfinancialrisks[J].StatisticsandProbabilityLetters, 2007, 77(14): 1522-1525. The uniformly asymptotic estimate for the tail probability of the sums of nonnegative and dependent random variables TANGFengqin1,2 (1. School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China;2. School of Mathematics Sciences, Huaibei Normal University, Huaibei 235000, China) Suppose that {Xi}i≥1isasequenceofnonnegativeandnon-identicallydistributedrandomvariableswhichbelongtothesubclassofheavy-taileddistributions-classC.ThemultivariatedistributionfunctionoftherandomvariablesisgovernedbytheFGMcopulafunction.Theuniformlyasymptoticestimateforthepartialsumsandrandomsumsofthesequence{Xi}i≥1arestudied,respectively.Theobtainedresultsextendthecorrespondingasymptoticestimateofthetailprobabilityofthedependentrandomvariables. precise large deviations; FGM; consistently varying 10.13471/j.cnki.acta.snus.2016.03.009 2015-11-30 安徽省高校自然科學(xué)研究一般資助項目(KJ2014B15) 唐風(fēng)琴(1983年生),女;研究方向:概率論極限理論;E-mail:tfq05@163.com O211.65;O A 0529-6579(2016)03-0055-042 主要結(jié)果的證明