The Law Of The Iterated Logarithm For Negatively Associated Random Variables

This paper has four parts, the first chapter is about the law of the iterated logarithm for negatively associated random variables with finite variance. The second chapter is about the law of the iterated logarithm for nonstationary negatively associated sequences. The third chapter is about the bounded law of the iterated logarithm for negatively associated random variables. The forth chapter is about the sufficient conditions for the NA random variables without stationary distribution to satisfy the law of iterated logarithm and the law of large numbers.The chief results of the first chapter are as follows:Theorem 1.1.1 Let {X_i, i > 1} be a strictly stationary negatively associated sequence with EX_i = 0, EX_i² < ∞, and σ² := EX₁²+20.ThenThe chief results of the second chapter are as follows: Theorem 2.1.1 Let {X_i} be a sequence of NA random variables and suffice the conditions of (2.1.1), (2.1.2)and (2.1.3), thenTheorem 2.1.2 Let {X_i} be a sequence of NA random variables and suffice the conditions of (2.1.1), (2.1.2)and (2.1.6),then there exits a constant ι ≥1, such thatTheorem 2.1.3 Let {Xi} be a sequence of NA random variables with i = 0,i > 1 and suffice the conditions of (2.1.12), (2.1.13),)and (2.1.14),then we obtain (2.1.5).Theorem 2.1.4 Let {Xi} be a sequence of NA random variables and suffice the conditions of (2.1.1), (2.1.12)and (2.1.13), furthemoreSn,m = X)^?+i' m>n - *> °2 = Hjï¿¡ï¿¡fm l supnt=lthen we obtain (2.1.11)and 0 < a1 < oo.The chief results of the third chapter are as follows: Theorem 3.1.1 If we have the conditions as follows: (1) There exits 2 < p < 3, such thatrn|* < oo;n=l(2)(BnL2(Bn))l> < on, n e iV(3) There exits /? > 0 ,such thatThen we have^1 < 8. a.s.Theorem 3.1.2 If there exits 3 < p < 4, such that EXl - 0, n > 1 and (3.1.1), assume that (3.1.2) and (3.1.3) hold, then we obtain (3.1.4).Theorem 3.1.3 Assuming thatp,n-ioo (In(1) If there exits 2 < p < 3, such that (3.1.1) holds, then we havelimsup !^1 < max(8, (2(1 + ^))*C), a.s.n-+oo <2n 3(2) If there exits 3 < p < 4, such that EXl = 0,n > 1 and (3.1.1) holds, assume that (3.1.2) and (3.1.3) hold, then we havelimsup >—^ < max(8,2C), a.s.n-?oo O,nCollary 3.1.1 Assume that lim Bn = oo, If on = (5nL2(A?))* suflBcen—foothe condition (3.1.1) of theorem 3.1.1, then we havelimsup------—^-----7- < 8, a.s.{BL{B))h ^Collary 3.1.2 If there exits 2 < p < 3(3 < p < 4, EXl = 0,n > 1) and 0 < pi < p, such thatsvvE\Xn\p < oo.liminfTT1^ > 0,n-+oon-+oothen we obtain (3.1.8).Collary 3.1.3 Assume that (nX2(an)):ron too, and there exits p > 2 ,such that\im(E\Xn\>)HnL2(al))*a? = 0,n—foothen we havelim — = 0, a.s. n-+ l,{on,n > 1} be a non-decrease sequence of positive numbers and lim an — +oo, {n*} C N suffice that for M > 1, we have Mank < ank+1 < M3ank+i,k = 1,2, ???, leto\ = ï¿¡ EXf, for some p > 2, we havei=nk+l^<+oc.Then we can obtain(1) for some A > 0, haven=l ukThen there exits a constant F > 0, such that lim sup ^ < F, a.s. (2) For any S > 0,then we have lim ^ = Oa.s..Theorem 4.1.2 Let {Xi} be a sequence of NA random variables withmean 0 and finite variance, Let Sn = f)Xj,s^ = ESl, n > 1, if we have the condition as follows:(1) there exit constants a > OandO < (3 < a, such thatsupE|Xn|2+a<+oo, liminf rTls^ > 0.n>l n^°°(2) there exits a constant K > 0, such that for all n G N, uniformly havelim —^--------= K,t=lthen we have\S I limsup tn 9r "... < l.o.s..