(m(2bn log logb)1/ =1) = 1 (4) Theorem 1 is a generalization of an author's result8 for the special case of symmetric. Bn, Bn+1 (2) bn 1=1

Transcription

1 ON THE LAW OF THE ITERATED LOGARITHM WITHOUT ASSUMPTIONS ABOUT THE EXISTENCE OF MOMENTS* BY VALENTIN V. PETROV LENINGRAD UNIVERSITY AND UNIVERSITY OF CALIFORNIA, BERKELEY Commuicaed by J. Neyma, February 19, 1968 Most of the kow results cocerig the law of the iterated logarithm ad its are obtaied for a sequece of idepedet radom variables geeralizatios which have fiite variaces or which are eve bouded. I the preset paper I shall prove two theorems o the law of the iterated logarithm without ay assumptios about the existece of momets of the radom variables cosidered. (1) Results.-Let { X}, = 1, 2,..., be a sequece of idepedet radom variables defied o a probability space (Q, F, P). Let us deote S = E Xk, V(X) = P(X < X). k =1 THEOREM 1. Suppose that the radom variables X1, X2,... have symmetric distributios. If there exists a sequece of umbers { B,} such that ad B t co, (1) B+1 (2) B, t2 sup fp(s < B21'x) - (27r>)1/2 fz e 2 dt = O[(log B)> ] x for some 5 > O (3) as o, the (m(2b log logb)1/ =1) = 1 (4) Theorem 1 is a geeralizatio of a author's result8 for the special case of symmetric distributios. I a previous paper" the assumptio was made about the fiiteess of the'variaces of the radom variables X1, X2,..., but there were o assumptios about the symmetry of distributios there. I order to verify whether the relatio (3) holds, we ca use estimates of the remaider i the cetral limit theorem give i the papers cited i refereces 1, 6, ad 7 without ay assumptios about the existece of momets. THEOREM 2. Let { a.}, I b}, ad { C} be sequeces of positive umbers such that a < c, c t o ad there exists lim(b/c) = -y, say. We set X if X < a, 0 if X,, aa. Let P(-1M E (yk b 1=1 E Yk) (5) 1068

2 VOL. 59, 1968 MATHEMATICS: V. V. PETROV 1069 If the Go z2 fl ax I2+ dv(x) < Ao, (6) P lim E (Xk- fjx <Ck xdvk(x)) = 7) = 1. (7) This theorem makes it possible to establish a iterated-logarithm-like relatio for the geeral case of ubouded radom variables if we are able to obtai a aalogous relatio for the trucated radom variables. I shall idicate some cosequeces of Theorem 2. Let { X,} be a sequece of idepedet radom variables havig zero meas ad fiite variaces. Let B = Ei EXk2 (8) k=1 ad B 0x. If there exists a sequece of positive umbers { A such that A = O[Bl/2(log log B) 1/2]) i - E iji >AkXIdVk(X) 0 (10) B k=1 as, ad E P( JX > A) < Ad (11) the (4) holds where B is defied by (8). A immediate cosequece of this last result is the followig theorem of Kolmogorov :4 if.x< m = o[b '/2(log log Bj) -1/2I, the (4) holds. -- It is possible to replace the coditio (11) by the weaker coditio (6) with a = A ad C2 = 2B log log B,, Theorem of Hartma ad Witer2 also follows from Theorem 2. (2) Proof of Theorem 1: Let us deote x() = (2B log log BO)'/2 for all sufficietly large. As a cosequece of the estimate (3) ad the asymptotic relatio t2 fxoce 2dt -x-le 2 (X O. c we have (log B,) -(1 + ps)b2 < P[S > bx()] < (log B,) - (12) for ay positive costats j& ad b < (1 + 8) '/ ad for all sufficietly large. It follows from the coditio (1) that for every T > 0 there exists a odecreasig sequece of itegers {kt} such that k -- o as k -- co ad Bk < (1 + r)k < Bk (k = 1,2,...) x2 (13)

3 1070 MATHEMATICS: V. V. PETROV PROC. N.A.S. (we assume that B. = 0). Takig ito accout the relatio (2), we coclude that Bk -' (1 + T)k (14) ad as k -A c. - Bk Bk- Il-" Bkr(1 + T)-1 From (12) ad (13) it follows that P[Sk> bx(k) I < (log B-b < [k log(l + T) I (15) for ay positive b < (1 + 5)1/2 ad for all sufficietly large k. Now we shall apply the iequality P(max Sk 2 x) < 2P(S. 2 x) which holds 1 <k < for symmetrically distributed idepedet radom variables X1,..., X,, ad for all x (Lo~ve, ref. 5, p. 247). As a cosequece of this iequality ad (15), we have E P [Sk > (1 + vy)x(k) I < (16) co for ay y > 0 wheres = max S. 1_<k; The followig iequalities hold for ay e > 0: P[S > (1 + E)x()i.oI < P[ max S > (1 + e)x(k-l)i -0o k- 1<<lc < P[Sk > (1 + E)X(tk-1)ijO. ]. I view of (2) ad (13), we have X(k)/X(k-1) < (1 + 2T)1" for all sufficietly large k ad therefore, P[S. > (1 + E)X()i.o.] < P[Sk > (1 + e)(1 + 2r) /'X(t)i.o.I. Let e be a arbitrary, fixed positive umber. Let y < e be aother positive costat. Fially, let the positive costat r be such that (1 + e) (1 + 2r) -I2 > 1 + -y. The - P[S > (1 + e)x(f)i.o.] < P[Sk > (1 + 'Y)X(k)i-o-.I Takig ito accout (16) ad applyig the Borel-Catelli lemma, we coclude P[S > (1 + e)x(r)i.o.i = 0 (17) for ay E > 0. Because of the symmetry of the distributios of the radom variables X1, X2,..., this result remais true if we replace here S by IS. I order to complete the proof of Theorem 1 it is sufficiet to show that P[S > (1 -E)X()i.o] = 1 (18) for ay e > 0. Let us deote 41(k) = [2(B7 - Bk1) log log (Bk - Bk ]12 From (14) we get log(bk-bk ) < log Bk < 2k log(l + T) for all sufficietly large k. As a cosequece of (14) we have also 4,(fk)/X(fk-1)

4 Tl2.l i a~zlb tic VOL. 59, 1968 MATHEMATICS: V. V. PETROV 1071 T~.1/1' ad B arc arbitcraryetiswehvi,(aii~ yrl{l'l\ I() - YA B") C 2 P(A) - P(BC) where BC is the complemet of B. Therefore, i view of (12), for ay positive y < 1 P [Sfk - Sk_ I> (1 -y)#(k)] ('vst'ts,w\e ililvve. S BS) = 1, -) >P( [Sk >( 1 - l)y1(k)] Slk I< 2 \(k) > P (Sk > (1-2)t&(k) - P(Sfkl. 2 X(k)) > (log Bk) );)( -y) _(log Bk _ 72T/5. The latter differece is greater tha C[k-('+")(1-a/) - k-2t/5] > -k- +,u)(l-y/2)' 2 for sufficietly large k ad r. Here C is a positive costat which does ot dcped o k. If we choose, sufficietly small so that (1 +,u)(l - ('Y/2))2 < 1, we obtai P [Sk Sk- I > (1 - y)#(k)] = o. Applyig the Borel-Catelli lemma oce more we have P[Sk - Sk I > (1 - -y)ji(k)i.-0] = 1 (19) for ay positive y < 1. Furthermore, (1-7))(k) - 2x(k-1) - [(1 - Y)T1/2(1 + T) - /2 2(1 + T)] (fl) as k -- o. From (17) we coclude that SS(c) < 2x() for > o(w) ad all w E Q except for a set of probability zero. If E is a arbitrary positive fixed umber, we ca choose positive umbers y ad r so that (1 - )')/2(1 ± T)/- 2(1 + ) '/2 > 1 - Takilg ito accout (19) we obtai P [Sfk > (1 - e)x(k)i.o.i > P [Sk > (1 - y)y,(k) -2x(,-)i.o. I > P[SIlk - Ski1> (1 - y)(k)i.o.] = 1. The equality (18) is a immediate cosequece of these estimates. is proved. Theorem 1 (3) Proof of Theoremt 2: Let J~ 0 if X < a, Z X if X > a, ad z _ X if a,, < IXI < C,1 t0 otherwise.

5 1072 MATHEMATICS: V. V. PETROV PROC. N.A.S. We have X =Y + Z ( = 1, 2, *.**.) =_ jz if IZ I<C Zfl{0 otherwise. Now we shall apply the followig result of Heyde:3 if {i } are idepedet radom variables, B} is a odecreasig sequece of positive umbers, B 00 ad if G if t,, < B,, 0 if > BX co 2 Z E Eo< 1 ~ 2+ B, the B-1 A (4 - EN) -O 0 with probability oe. Accordig to this theorem k = I we have We fid C-j _d (Zk - EZk) 0 with probability oe. (20) k I Xk fix, - < c, xdvk(x) = Yk- EYk + Zk - EZk- From (5) ad (20) it follows that the relatio (7) holds. The author thaks Professor C. C. Heyde for sedig him his mauscript' before publicatio, ad Professor V. Strasse for his helpful commets. * This paper was writte while the author was a participat i the Soviet-America scietific exchage program. 1 Berry, A. C., "The accuracy of the Gaussia approximatio to the sum of idepedet variates," Tras. Am. Math. Soc., 49, (1941). 2 Hartma, P., ad A. Witer, "O the law of the iterated logarithm," Am. J. Math., 63, (1941). 3 Heyde, C. C., "O almost sure covergece for sums of idepedet radom variables," Sakhya, i press. 4 Kolmogorov, A. N., "Ueber das Gesetz des iterierte Logarithmus," Math. A., 101, (1929). 5 Lobve, M., Probability Theory (Priceto, N.J.: Va Nostrad Co., 1963), 3rd ed. 6 Osipov, L. V., ad V. V. Petrov, "O the estimatio of the remaider i the cetral limit theorem," Theor. Probability Appl., 12, i press. 7Petrov, V. V., "Cetral limit theorem without the hypothesis of the existece of momets," i Limit Theorems Statistical Iferece (Tashket: Izdat. "Fa," 1966), pp (i Russia). 8 Petrov, V. V., "O a relatio betwee a estimate of the remaider i the cetral limit theorem ad the law of the iterated logarithm," Theor. Probability Appl., 11, (1966).