Set Of Columns Of The Random Variable Limit Theorem

This thesis is finished under the guidance of my tutor, professor Zhang Lixin, during my master of science. It consists of two chapters:Chapter 1 On complete convergence for arrays of rowwise negatively associated random variables.Because of its applications in multivariate statistical analysis, reliability theory and osmosis theory, negatively associated random variables' applications in many fields become more and more, such as oceanics, bioengineering, meteorologic engineering, environmental engineering and iatrology etc. So its convergence properties have drawn many attentions from scholars. Matula have studied the Kolmogorov inequality and three series theorem for negatively associated random variables in 1992. Petrov have extended the lemma Borel-Cantelli to the case of negatively associated random variables in 1991.For arrays of rowwise independent random variables, Gut(1992), Wang et al. (1993), Hu et al.(2003) and Kuczmaszewska(2004) have studied their properties of complete convergence. Sung et al. (2005) got a correspondingly common form.Theorem 1.1 Let{X_ni,1≤i≤k_n,n ≥ 1} be an array of rowwise independent random variables and {α_n,n ≥ 1} a sequence of positive constants such thatSuppose that for every ε > 0 and some δ > 0:(ii) there exisits J≥ 2 such thatThen^fliX !) foralUX).n=l i=lBy extending this conclusion, we get a complete convergence theorem for arrays of rowwise negatively associated random variables. Theorem 1.2 Let{Xnj,\ < i < kn,n > 1} be an array of rowwise negatively associatedrandom variables and{an,? > 1} a sequence of positive constants such that> a = oo.Suppose that (i),(ii) and (iii) in Theorem 1.1 are satisfied. Then,/(! IX i>s) < oo for all s > 0.n=l 1=1Chapter 2 The weak law of large numbers for arrays of random variables in martingale type/;Banach spaces.Relative to classical weak law of large numbers, Gut(1992), Hong et al.(1995), Hong et al.(1996), Sung(1998) and Sung(2005) have studied the WLLN for arrays of random variables. Hong(1996), Adler et al.(1997), Hong et al.(2000) and Ahmed et al.(2002) have extended the WLLN for arrays of random variables to the case of martingale type/? Banach space valued random variables. Hong et al.(2000) get:Theorem 2.1 Let {Vnj., j > 1, n > 1} be an array of random variables in a real separable,martingale type p(l < p < 2)Banach space, and{Nn,n > 1} be a sequence of positive integer-valued random variables such that for some nonrandom sequence of positive integers &M -? oo ,we haveLet {aw■, j > 1, n > 1} be an array of constants and let/(?) = 1 / max | anj \ satisfyingasSuppose that there exists a positive nondecreasing sequence{g(m),m>0},g(0) = 0,such thatasm=\ mSuppose the uniform Cesaro-type conditionholds.Then we have WLLN/VO)^^ as(M? * = a(Vni,\\,n>\ ,and FnO={0,n},n>\.By extending Theorem2.1,we getTheorem 2.2 Let {X^, un < i < vn, n > 1} be an array of random variables in a realseparable, martingale type p (1 < p < 2) Banach space, where {un > -oo,? > 1} and{vn < +oo, n > 1} are sequences of intergers.Let{&n,?>l}be a sequence of positive integers and{anj,un1 \Ssupsup—￡aPQ\ X^ ||> g(a)) < qo ,a>0 n>l /Cn ,_uand lim sup—V aP(\ | Xni \ \> g (a)) = 0.holds. ThenV^ ani (Xni - cni) ->0 in probability. Where cni = E(XJm_||St^)} | F ?_M), F m = cr(ZB>, 1 <;< /),/>!,?>! ,...