【摘要】 设{Xn}是概率空间(Ω,Γ,P)上的随机向量序列,I是{Xn}对应经验过程的可测函数指标集合.有关函数指标集合上经验过程的概率指数不等式研究一直被条件supf∈If≤M,M>0所限制,以致对应的经验过程不能包含象样本均值和样本方差等常用的统计量,这使经验过程概率指数不等式的应用受到了极大的限制.记L4(P)={f:Ef4(Xi)<∞,i≥1,f(.)是Borel可测的实函数},在IL4(P)和独立不同分布的样本条件下,无界函数指标集I上经验过程的概率指数不等式被研究.利用一种新的对称化思想和一种新的截割概率空间的方法,无界函数指标集I上经验过程的概率指数不等式被给出.这些不等式与定理Ⅱ.33[1]以及文献[2~4]等中关于有界函数指标集上经验过程的概率指数不等式有着本质的区别.更多还原

【Abstract】 Let {X-（n）}be a sequence of independent random vectors defined on a probability space（Ω,,P） valued in a measurable space（,A） and X-（i） have the law P-（（i））,i1. Denote by P-（n） the empiricl probability measure of X-（i）,1in. Suppose that I is a class of measurable real functions on （,A）.Define Qf:=fdQ for any probability measure Q and any fI. Let -（n）:=1/nⁿ-（i=1） P-（（i）） and E-（n）（）=n（P-（n）（）--（n）（））. {E-（n）f:fI},n1,is called an empirical process over I. Empirical processes are extensively applied to many fields,such as economics,natural science,social science etc. The famous Markowitz portfolio selection model^[5] is set up on the basis of expectations and variances. Because sample means and sample variances can be written in the forms of empirical processes,it is sufficient to illustrate the importance of empirical processes in applications. It is very essential to establish probability exponential inequalities of empirical processes in order to study the asymptotic behaviours of empirical processes. For the cases of i.i.d.random samples, the examples can be found in and references therein. Under the conditions of independent （possible non-i.i.d.） random samples, Alexander and Talagrand^[4] give the probability exponential inequalities of empirical process over I. However,up to now,the study on probability exponential inequalities of empirical processes over I has been restricted by the condition of sup-（fI）fM（M>0） so that the corresponding empirical processes cannot cover many useful statistics like sample means and sample variances. Let L⁴（P）={f:Ef⁴（X-（i））<,i1,f（） is a Borel measurable real function}. The main object of this paper is to establish the inequality of empirical processes over unbounded classes of functions when IL⁴（P）. In order to establish these inequalities,a new symmetrization idea is proposed and a new truncating method is used. Under the condition of independent （possible non-i.i.d.） random samples, this paper presents the probability exponential inequalities of empirical processes over unbounded classes of functions with the 4th order monent. There is entitative difference between the probability exponential inequalities of this paper and that over bounded classes of functions in Theorem .33.Main ResultsUnder the conditions of IL⁴（P） and independent （possible non-i.i.d.） samples,we will give the probability exponential inequalities of the empirical process over an unbounded class of functions.Let （I,d） be a pseudo-metric space, the covering number is defined as follows.N（u,d,I）=min{m:there are f-（1）,...,f-（m）I such that（sup）fI（min）1jmd（f,f-（j））u}for any u>0,（1）see ref.（[1]） （p.143）.Denote-（I）:=sup-（fI）,d-（n,r）（f,g）:=(P-（n）f-g^r)^1/r and -（n,r）（f,g）:=(-（（n））f-g^r)^1/r for any f,gI,r=1,2,n1. Let N-（r）（u,P-（n）,I） and N-（r）（u,-（（n））,I） respectively stand for the random covering numbers of （I,d-（n,r）） and （I,-（n,r））,r=1,2. The following condition is useful.Condition 1^[7]There exist constants A>0 and W>0 for which N-（2）（u,P-（n）,I）Au^-W and N-（2）（u,-（（n））,I）Au^-W for any u(0,1] and any n1.Dedinition 1^[1]Suppose that F（） is a non-negative real function with fF for any fI. F is called an envelop function of I.In order to assure the measurability of P-（n）f--（（n））f-（I）, we need to use the condition of "permissible" and its definiton can be found in Appendix C of [1].Theorem 1.1. Assume definition I is a permissible class of functions with an envelop function F>0 and sup-（i1）EF⁴（X-（i））<. Furthermore suppose that 0<lim inf-（n）1nⁿ-（i=1） Var（f（X-（i）））-（I） and Condition 1 holds. For any （0,1/2）,there are an integer N>0 and a measurable set S with P（S）>1-,if nmax{N,16V²-（n,I）/ε²} for any ε(0,1],then P-（S）（P-（n）f--（（n））f-I>ε）<16（512）^WA²（1/ε）^2Wexp{-nε²/(258V²-（n,I）)}（2）where P-S:=P（S）/P（S） and V-（n,I）=1nⁿ-（i=1） Var（f（X-（i）））-（I）^1/2.Remark 1. Noting that f（）-（fI） is an envelop function of I, if I is permissible, then there exists an envelop function of I, F>0,to satisfy sup-（i1）EF⁴（X-（i））< if and only if sup-（i1）E(f（X-（i））⁴-（I）)<. Thus, under the conditions in Theorem 1.1,there exist E[f⁴（X-（i））],for any fI and i1, and they are uniformly bounded.Remark 2. If Conditon 1 is satisfied, then, for any ε>1 and n1, N-（2）（ε,P-（n）,I）A and N-（2）（ε,-（（n））,I）A. Therefore, if ε（162）,then （2） in Theorem 1.1 can be rewritten as follows.P-（S）（P-（n）f--（（n））f-（n）-（I）>ε）<16A²exp{-nε²/(258V²-（n,I）)}（3）See （27） and the proof of Theorem 1.1. Corollary 1. If the conditions of Theorem 1.1 are satisfied, there exists a constant L>0 such that （lim sup）nnP-（n）f--（（n））f-（I）/(V-（n,I）（logn）^1/2)L a.s.（4）Corollary 1 implies that the methods and results of the paper are very effective in study of the asymptotic properties of empirical processes over unbounded classes of functions.In order to illustrate this fact furthermore,we give an example of application.Example 1.1. Let {X-（n）} be a sequence of independent（possible non-i.i.d） random variables on a probability space （Ω,,P） valued in R with sup-（i1）E[X⁴-（i）]<.Set ²-（n）=1/nⁿ-（i=1）Var（X-（i））,i1. If 0<lim inf-（n）²-（n）,then,for any （0,1/2）, there exist an integer N>0 and a measurable set S with P（S）>1- such that, if n max{N,8²-（n）/ε²} for any ε(0,1],thenP-（S）(1/nni=1（X-（i）-EX-（i）>ε）<8 exp{-nε²/(65²-（n）)}（5）The result of Example 1.1 is obvious from Theorem 1.1 and its proof process.In fact,taking I={f（）},f（x）=x,xR in Theorem 1.1,because the covering numbers of I equal 1,（5） can be obtained by changing some coefficients in the proof of Theorem 1.1.Remark. In the last example,the restriction that {X-（n）} is uniformly bounded in the famous Hoeffiding inequality^[6] is successfully deleted.更多还原