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(1)Probability Inequalities of Exponential Type for Continuous Parameter Processes and their Applications By Hiroshi NEGISHI* 1. Introduction.. x ///. Let {Xt, -oo<t<oo} be a continuous parameter strict!y stationary process consisting of uniformly bounded random variables. Let vet be the a-field. generatedby{X,,-oo<s.<.t}. . ･ . Suppose that for a real valued function g(s) (ev(s)) such that g(s) l O (cr(s) t O). (1.1) IE(Xel.ce-s)-rE(Xe)IS-q(s) (a.s.). '. '. or --. (1.2) EIE("XolS-s)rE(Xo)[ lll a(s)･ In [4], the author has given an exponential bound of the probability for the sum of random variables satisfying (1.1) or (1.2) with the discrete parameter.. Our results in this paper are obtained by the same method as that of [4]. Theorem 1 and Theorem 3 are some extension of those in [1]. Examples satisfying (1.1) are presented in [1]. If {Xt} is a strictly station-. ary strong mixing process of uniforrnly bounded random variables, then (1.2) is satisfied. (cf. [2]). 2. Theorems.. } --. p. i. Let {Xt, -oo <t<oo} be a continuous parameter strictly stationary process consisting of random variables such as to be uniformly bounded and measurable in the sense of [3]. It follows from a theorem of [3] (Ch. II, Theorem 2.7) that we can define the random variable l",iXtd4 (ti<t2). Moreover, 'E(Si'iXtdt)==. .t. sE(Xt)dt holds. t2. tl. THEoREM 1. SuPPose that {Xt, -oo<t<oo} is a strictly stationaror measzarable Process with P(]X,1:;ll)=1 and E(X,)=pt. I.f this Process satisfies (1.2) with a ftenction cr(s), then for any x>O and any Positive integer m. '. ' " Department of Mathematics, Faculty of Education, Yokohama National University.. .. t ttt.

(2) 18 H. NEGIsHI (2.1) P(iS,TXtdt-Tpt>x);l2exp[-x2/(72m2T)]+(3/x)Tcr(m). THEoREM 2. SuPPose that {Xt, -oo<t<oo} is a strictly stationary measurable Process with P(IXtl;Sl)=1 and E(Xt)=pt. If this Process satisy7es (1.1) with a function g(s), then for any x>O and any Positive integer m,. (2.2) P(S,TXtdt-Tpt>x):S2exp[---x2/(16m2T)+(x/(8m2))g(m)].. PROOF OF THEOREM 1. Define. (2.3) 6,=Sl-,X,dt (1==1,2,･･･). li. ?AneCo?eim't ftOhla12V¥grfraOnMy tsh>eodefinitiOn Of the conditional expectation and FubiniJs. 1. ' E(Sll-,Xedt･S)j-s)=Sl.-,E(Xe-sj-.s)dt (a.s.), we have from (1.2) (2.4) ElE(6,i.s,-.,)-ptl==ESI.,[E(X,-ptl.S,-,)]dt ;I$jl-,ElE(Xt-pt1.ce,-.,)ldt==Sl-,EIE[E(X,--ptI.s,.-,)l.f{B,-,]ldt. ;lllSl-,EiE(Xt-'jaI-!{Bt-s)ldt:;Iila(s).. If [T] is the largest integerSZ we have. (2.s) S,Tx,dt==SiT]X,dt+S,.,,,,.Xtdt '. ==[T] 2 8j+eT )'=1. where. (2･6) ieTl=S,.,,,,,.,Xtdti:Sl (withprobabilityl).. l'. / t'. ' Noticing (2.4), we can apply the technique of [4] to [zT] ej. Denote for any. d=:1. positive integer m. Xji=E(6jl.mj..i)-E(6jl.S)j--i-i). a"d. yj.=E(el.s?j-.)-pt (i'=i,･･･[T],i--i,･･･m)･. Then it is easily seen that. [T] [T]7n-1 CT] (2･7) }.i]=,(C?j-'iet)== P., i=,Xji+ ,l=, Yjm=ST+S; (say)･. ,.

(3) Probability lnequalities of Exponential Type 19 From Lemma of [4], it is seen that for each i {X,･i} is a multiplicative sequence. Therefore, from the same argument as [4], we obtain for any real u. (2.8) Eexp (uST) S. exp (2u2m2T). Hence, from (2.8) and the Markov inequality. (2.9) P(iS.I>x/3)=P(S.<-x/3)+P(S.>t/3) ;;l exp (- ux/3)[E exp (uST) + E exp (-- uST)]. ;S2exp(-ux/3+2u2m2T). Choosing u==x/(12m2T) in (2.9), we obtain. (2.10) P(I STI>x/3) ;:E2 exp (- x2/(72m2T)).. K `. On the other hand, it follows from (2.4) that. (2.11) P(IS;I>x/3)i:S(3/x)EIS;1S(3/x)cr(m)T, Si"Ce. sgx,dt-Tpt..s,l+-s;-16.+,,,(T-[T]),' t'･''r'" -, 1,/. we have from (2.6), (2.10) and (2.11). .i)( !gX,dt-T?ee >x);:$P(IS.I>x/3)+P(1S*.I>x/3). +P(IsT+pt(T-[T])l>x/3) ;$2exp(-x2/(72m2T))+(3/x)a(m)T. because, since IeT+pt(T--[T])l>2, ･ P(leT+pt(T-[T])I>x/3)=O for x>6. Thus, the proof is completed.. PRoOF oF THEOREM 2. Since, it follows from (1.1) that for s>O. j 1E(6,I.{B,-,)-ptl;SiS,.-,lE(X,-ptI.ce,..,)Idt. K. j. ==Sll-,lE[E(Xt-ptl.s,H,)]s,･n,]Idt. '. :ISISl'-.,lE(X,-pt1.ca,-,)ldt;ISg(s). the proof is in the same line as that of Theorem 2 in [4] and so is omitted 3. The large deviation of the empirical distribution function.. Let F denote the one dimensional marginal distribution function of a strictly stationary measurable stochastic process {Xt, -oo<t<oo}. Define. FT(y)=(1/T)S,TI,(Xt)dt (-oo<y<oo). '.

(4) 20 , H.'NEGIslHI where I, is the indicator function of the interval (-oo, y]. Then FT is the empirical distribution function based on {Xt,O,<..t=<.T}. ' ,. Now, suppose that for everyy ' (3.1) ElE(I,(Xt)1.E{Bt-,)-F(N)1;:i;a(s)iO (s--Foo), THEoREM 3. (of. Theorem6 in [1]) Let {Xt,-oo<t<oo} be a strictly stationa7y measurable stochastic Process with one dimensional marginal distribution function F and satistv (3.1). Illf a(s)==O(e-2S) for some 2>O, then forany e>O there is a Positive constant C, dePending on e and 2, such that. (3･2) P(supIFT(y)-FO,)l>e);Sexp(-CTii3). '. y PRooF. Let m=[Tii3]. From Theorem 1, we obtain for each y. (3･3) P(IFT(y)-F(y)l>e);:S2exp(-E2Ti'3/72)+(3/e)exp(-RT'i3) $Coexp(-Z,Tii3) where Co=max (2, 3/e) and Zo==min(e2/72, R). In the same way as [1], (3.3) is sufficinent to obtain (3.2).. References. '. [1] BHATTAcHARyA, P.K., Probabilities of large deviations of sums ofrandomvariables. Sankhy-a, Series A, 34, 9-16 (1972).. [2] DvoRETzKy, A,, Asymptotic normality for sums of dependent random variables. Proc. 6th Berkeley Sympos. Math, Statist. Probab. vol, 2, 513-539 (1970).. [3] DooB, J,L., Stochastic Processes. John Wiley & Sons, New York, 1953. [4] NEGisHi, H., Probability inequalities for exponential type for sums of weakly. .dependentrandomvariablesandtheirapplications.(submitted) ･ '. { g.

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