Mathematica Slovaca Detlef Plachky A versio of the strog law of large umbers uiversal uder mappigs Mathematica Slovaca, Vol. 49 (1999), No. 2, 229--233 Persistet URL: http://dml.cz/dmlcz/131004 Terms of use: Mathematical Istitute of the Slovak Academy of Scieces, 1999 Istitute of Mathematics of the Academy of Scieces of the Czech Republic provides access to digitized documets strictly for persoal use. Each copy of ay part of this documet must cotai these Terms of use. This paper has bee digitized, optimized for electroic delivery ad stamped with digital sigature withi the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz
rvbthematica Slovaca 1999 Math. Slovaca, 49 (1999), No. 2. 229-233 si^sesís.5 i?, tílí. u ««A VERSION OF THE STRONG LAW OF LARGE NUMBERS UNIVERSAL UNDER MAPPINGS DETLEF PLACHKY (Commuicated by Miloslav Duchoň) ABSTRACT. Let (ft, A t P) stad for some probability space, 0 for a separable topological space, ad (Y, y) for a measurable space. Furthermore, /: Y x 0 > R is some fuctio such that /^ is y -measurable for all d G 0 ad {/ : y G y} is poitwise equicotiuous. It is proved that for ay sequece X lt X 2i... of Y-valued radom variables, which is i.i.d. relative to P such that E(\f(X lt 'd)\) < oo is valid for ay # G 0, there exists some P-zero set N satisfyig f(xi(w),0) -» E(f(X lt #)) t u G ft \ N, for all 0 G 0. This result is i=l illustrated by examples ad compared with kow uiform versios of the SLLN. 1. Itroductio ad mai result Let (fl,.4,p) be a probability space, (Y, y) a measurable space, ad /: Y x 6 -» R a fuctio such that f# is ^-measurable for all i? G 0, where 6 stads for some o-empty ad ot ecessarily coutable set. The it seems quite iterestig to iquire, whether the followig uiform versio of the strog law of large umbers (SLLN) holds true: Does there exists for ay (w.r.t. P) idepedet ad idetically distributed (i.i.d.) sequece of Y-valued radom variables X 1,X 2,... satisfyig E(\f(X 1, f d)\) < oo, d G 0, some P-zero set N G A such that lim E /(*»M,tf) -> E(f(X 1} #)) il\n ad ay d G 0? Now it will be show that the followig coditios are sufficiet: holds true for all u G 1. 0 is some separable topological space. 2. /: Y x 0 - > R has the property that {/ : y G Y} is poitwise equicotiuous. AMS Subject Classificatio (1991): Primary 60A10, 60B12. Key words: Uiform strog law of large umbers. 229
DETLEF PLACHKY The poitwise equicotiuity of {/ : y G Y} implies for ay tf 0 G 0 ad e > 0 the existece of some eighbourhood L7 (I? 0) satisfyig f{y^0) e < f(y,fl) < f(y^o) + > ^ e ^e(^o)'! / ^ i fr m which the iequalities /W"M 0 )-e< E/(-W>*) < ^ /(^M^o)+^ "e, 2=1 2=1 2=1 i? G U (i? 0 ), follow. Therefore, the iequalities limsupi^/(^h^o)- < l i m s u P^E/(^H^) >oo Tl. л юo Tl. л _ v г=l -+oo г=l 2=1 1 " < limsu >oo Tl. p-e/( x >)><>+e are valid for all u G f) ad ay tf G t7 (i? 0 ), i.e. limsup^ E f{x i (uj),'d) I ->oo i=l limsup^ E /(^i( a; )?^o) <, tj G fi, i? G c7 (i? 0 ), holds true, which proves >oo 2=1 that the fuctio defied by d -> limsup^ E /O^iv^)?^) > $ 0, is coti >oo i=l uous for all u G fi. By a similar argumet the fuctio itroduced by i? > limif ^ E fi^ii^)^)» $ G 0, is cotiuous for all u; G -1. Furthermore, the ->oo i=1 fuctio i? -> E[f(X 1^)), i9 G 0, is cotiuous. Now the classical SLLN imf plies that the set S itroduced by (w,i))eflx0: limsup E /(^im' ^) < ^ -»oo 2=1 E(f(X^)) or limifis/^m.tf) > E(f(X lt 0))} satisfies P(5 tf ) = 0 for all d -»oo i=1 ) G 0. Furthermore, the cotiuity of the fuctios d -> limsuple/(^w^), * "> limifi /(^), ad i? -> E(f(X 1,^)), ->oo 2=1 ->oo ^=1 i? G 0, together with the existece of some coutable ad dese subset 0' of 0 yields the uiversal P-zero set TV G A defied by IJ S# of the type described by the followig theorem. THEOREM. Let (fi,*4, P) deote a probability space, 0 some separable topological space, ad (Y, y) some measurable space. Furthermore, let f: Y x 0 -> R. 6e a fuctio such that f# is y-measurable for ay i)g0, ad {/ : y G Y} 25 poitwise equicotiuous. The for ay sequece X i : f) > Y, z = 1,2,..., o/ y-measurable radom variables, which are i.i.d. with respect to P ad satisfy E[\f(X 1,, d)\) < co, i? G 0, there exists some P-zero set N G A such that lim 1 E/(X-(a;),i?) = ^(/(X^i/)) is valid for all u <E tt\n ad ay tie @. i=i 230 2 = 1
THE STRONG LAW OF LARGE NUMBERS UNIVERSAL UNDER MAPPINGS 2. Examples ad compariso to kow results The followig first example shows that oe caot drop the assumptio that : {f y y e Y} is poitwise equicotiuous ad that the correspodig domai O is a separable topological space without itroducig some other coditios. EXAMPLE 1. (Projectios of radom vectors) First of all it will be show that the exceptioal zero-set occurrig i the SLLN caot be empty i geeral. For this purpose let f2 stad for the iterval [0,1) with the correspodig Borel a-algebra B([0,1)) ad let X k : fi > R stad for the B([0,1))-measurable radom variable deed by X k (u) = u k, u G ft, oo k G N, where u = ^ T, u k G {0,1}, k G N, is the dyadic expasio k=i of u, which is uique if there does ot exist ay j GN satisfyig u k = 1, k > j. The the radom variables X X,X 2,... are idepedet ad idetically distributed with respect to the probability measure P o B([0,1)) itroduced as the Lebesgue-measure restricted to #([0,1)). Obviously, the set N defied by iu G [0,1) : lim "i+-+" ^ l) is ot empty. Now let the set Y stad for [0,1] T, T beig some ucoutable set, where the cr-algebra y of subsets of Y is itroduced as the direct product A t. Here the cr-algebra A t of subsets of [0,1] coicides with B([0,1]) for all. The cr-algebra y has the followig property: For ay A G y there exists some coutable subset S of T such that (yt) A ad y s = y' s, s e S, for some (y' t ) G Y implies (y' t ) G A, i.e. the coutable subset S of T determies A. Now if oe itroduces Q by the set cosistig of all oe-dimesioal projectios 7r t : [0,1] T > [0,1],, ad the fuctio /: Y x 6 -> R by f((y t ), t ) = y t = t ((y t ) ), t et, the f is ^-measurable for all t G T. Furthermore, i coectio with the Y-valued ad y-measurable radom vectors Y : [0,1) T r [0,1] T defied by Y ((u t ) ) = (X (u t )), e N, where X, G N, has bee itroduced at the begiig of this example, oe gets that the exceptioal zero-sets N s deed by Uu t ) G [0, if : lim ^(X^uJ + + X (w,)) ^ ^} is equal to X A t, A t = [0,1), t e T \ {s}, A s = N. Here TV has already bee defied at the begiig of Example 1 ad the uderlyig probability measure o 0 A t, A t = B([0,1)), t e T, is the direct product (g> P t, P t = P, t G T, P beig the Lebesgue-measure restricted to B([0,1)). Now it will be show that for IJ N t there does ot exist ay M e (g) A t, A t = B([0,1)),, satisfyig ( (g) P t ) (M) = 0 ad IJ N t C M. For this purpose oe observes MGT ' 231
DETLEF PLACHKY T that the iclusio IJ N t C M together with some (u t ) tgt e [0, l) results i fat) ^ M r, i.e. M = [0,1) T, which might be see as follows: Let S deote some coutable subset of T, which determies M ad let fa' t ) be ay elemet of [0,1) T satisfyig u' t = u t, t e T \ {t 0 }, ad u' t e iv, where t 0 is some elemet of T\ S. Hece (u;j) tgt G JV to together with N t c M implies fat) e M - Fially, J N t ad X B t, where B t stads for /V c, * G T, are t6t disjoit, i.e. 1 1 ^ ^ ^ ^ = B ([ > X ))> t e T > holds true - The secod example results i some applicatio of the precedig theorem. EXAMPLE 2. (Power series with radom coefficiets) Let Y ad 0 stad for secod coutable topological spaces ad let /: YxQ -» R be some cotiuous fuctio with respect to the correspodig product topology of Y x 6. The there exists for ay y ey some eighborhood U(y) such that {/ / : y' C U(y)} is poitwise equicotiuous (sice otherwise there would exist tf 0 e 0, y Q e Y, ad e Q > 0 satisfyig /(y > # ) - /(y,# 0 ) > 0 > G N, where (y ) GN, y G Y, G N, ad (# ) N, tf G 6, G N, are sequeces with lim y = y Q ad lim ti = tf Q, which is a cotradictio to the >oo >oo property of / to be cotiuous) ad a theorem of Lidelof (cf. [2; 1.4.13, p. 12]) yields the existece of some coutable collectio U(y k ), k = 1,2,..., satisfyig oo U U(y k ) = IJ U(y) = Y. Now the theorem above results i the existece of k=l yey some uiversal zero set with respect to {/ : y G Y} i coectio with the SLLN, if the cr-algebra y of subsets of Y is chose as the correspodig Borel Ii5l cr-algebra B(Y). I particular, i coectio with ]T) l a JVr <» 1^1 < ^o =l for some tf 0 > 0 ad some (a ) GN ~* ^N > oe m^&^ itroduce the cotiuous fuctio /: Y x 6 -> R with Y = {(y ) en G R* : \y \ < \a \, G N}, oo ad O = (-tf 0> i? 0 ) deed by /((i/ ) en,<>) = E ^, (» ) e N r, tf 0, =l where R N is equipped with the product topology ad 6 with the relative topology of R. Remark. (Compariso with kow uiform strog laws of large umbers) I [3; p. 107-111] ad [5; p. 854] oe might fid the followig uiform versio of the Strog law of large umbers: P{ lim sup /(*.,*)--E(/(*i,*)) =0} = 1 I-»OO0 G0 i=1 J uder the assumptio that 0 is some compact ad metric space (tacit assumptio, cf. [3; p. 110]), ti -> /(y,tf), 1? G 0, is cotiuous for all y G Y, ad there 232
THE STRONG LAW OF LARGE NUMBERS UNIVERSAL UNDER MAPPINGS exists some y -measurable fuctio g: Y -> R such that gox x is P-itegrable ad /(y, tf) < g(y), y Y, i? E 0. This result might also be derived easily by the theorem above together with a versio of the theorem of Arzela-Ascoli, which might be foud i [6; p. 369]. However, there appears the stroger poitwise equicotiuity assumptio for {/ : y ey}. Fially, oe might cosult [1; p. 4], ad [4; p. 1308], for a versio cocerig the existece of some uiversal P-zero set i coectio with the SLLN. Ackowledgemet I would like to thak the referees for critical commets improvig the represetatio ad Professor Z. A r t s t e i for drawig my attetio to the problem cocerig a uiversal versio of the SLLN with respect to mappigs. REFERENCES [1] ARTSTEIN, Z. WETS, R. J.-B.: Cosistecy of miimizers ad the SLLN for stochastic programs, J. Covex Aal. 2 (1995), 1-17. [2] DUNFORD, N. SCHWARTZ, J.: Liear Operators I, Iteгsciece, New York, 1964. [3] FERGUSON, T. S.: A Course i Large Sample Theory, Chapma & Hall, Lodo, 1996. [4] HESS, CHR.: Epi-covergece of sequeces of ormal itegrads ad strog cosistecy of the maximum likelihood estimator, A. Statist. 24 (1996), 1298-1315. [5] JOHNSON, R. A.: Asymptotic expasios associated with posterior distributios, A. Math. Statistics 41 (1970), 851-864. [6] RUDIN, W.: Fuctioal Aalysis, Tata McGгaw-HШ Publishig Compay, New Delhi, 1982. Received September 19, 1994 Revised Jue 9, 1997 Istitute of Mathematical Statistics Uiversity of Muster Eisteistr. 62 D-48149 Muster GERMANY 233