ARTICLE IN PRESS
Statistics & Probability Letters 76 (2006) 503–506 www.elsevier.com/locate/stapro
A converse to precise asymptotic results Deli Lia,,1, Aurel Spa˘tarub,2 a
Department of Mathematical Sciences, Lakehead University, 955 Oliver Road, Thunder Bay, ON, Canada P7B 5E1 Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Calea 13 Septembrie No. 13, 761 00 Bucharest 5, Romania
b
Received 23 February 2004; received in revised form 25 July 2005 Available online 1 September 2005
Abstract P Let fX ; X n ; nX1g be i.i.d. random variables with partial sums fSn ; nX1g, put f ðÞ ¼ n an PðjSn jXbn Þ; X0, and assume 2 2 there exist functions g and h, such that lim&0 gðÞf ðÞ ¼ hðEX Þ whenever EX o1 and EX ¼ 0. We prove the converse result, namely that lim sup&0 gðÞf ðÞo1 and bn ¼ OðnÞ imply EX 2 o1 and EX ¼ 0. r 2005 Elsevier B.V. All rights reserved. MSC: primary 60G50; secondary 60F15 Keywords: Tail probabilities of sums of i.i.d. random variables; Precise asymptotics; Symmetrization
1. Introduction and result Let fX ; X n ; nX1g be i.i.d. random variables with PðX a0Þ40 and partial sums fS n ; nX1g, and consider series of the type X f ðÞ ¼ an PðjSn jXbn Þ; 40, n
P where an ; bn 40 and n an ¼ 1. Then there exists a threshold a, such that f ðÞ ¼ 1 for oa, while f ðÞo1 for 4a. The so-called precise asymptotic problem consists in finding, under appropriate moment conditions, an elementary function gðÞ40; 4a, such that lim&a gðÞ ¼ 0 and lim&a gðÞf ðÞ ¼ la0; 1, i.e., in establishing that f ðÞl=gðÞ as & a. For almost exhaustive references on this area, see Spa˘taru (2004a,b). Except for the cases, where X is assumed to belong to the domain of attraction of a stable law, most known such asymptotics are derived by assuming at least EX 2 o1, and then it turns out that l ¼ hðEX 2 Þ for some finite function h. Typical precise asymptotic results suited to have genuine converses are those where the assumption ‘‘EX 2 o1 and EX ¼ 0’’ is stronger than ‘‘f ðÞo1 for some 40’’. When a ¼ 0, such a general result is as follows. Corresponding author.
E-mail addresses:
[email protected] (D. Li),
[email protected] (A. Spa˘taru). Supported by a grant from the Natural Sciences and Engineering Research Council of Canada. 2 Supported in part by the Ministry of Education and Research of Romania under Grant CNCSIS 27639. 1
0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.08.017
ARTICLE IN PRESS D. Li, A. Spa˘taru / Statistics & Probability Letters 76 (2006) 503–506
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Theorem A. There exist positive and finite functions g and h on ð0; 1Þ with limx!1 hðxÞ ¼ 1, depending solely on the sequences fan g and fbn g, such that lim gðÞf ðÞ ¼ hðEX 2 Þ, &0
whenever EX 2 o1 and EX ¼ 0. We obtain the next converse to Theorem A. Theorem 1. Let g be as in Theorem A, and assume that X an PðjSn jXbn Þo1, lim sup gðÞ &0
(1)
n
where bn ¼ OðnÞ. Then EX 2 o1 and EX ¼ 0. Remark 1. (1) implies that f ðÞo1 for all 40. If this, in turn, implied that EX 2 o1 and EX ¼ 0, there would be nothing more to prove. The two examples below are veritable converses to important precise asymptotic theorems. Proof. We first show that (1) implies EX 2 o1. To do this, let fX 0 ; X 0n ; nX1g be an independent copy of fX ; X n ; nX1g, and consider the symmetrized random variables Y ¼ X X 0 ; Y n ¼ X n X 0n , U n ¼ Y 1 þ þ Y n , nX1. Next, for any l40, write Y n ðlÞ ¼ Y n IfjY n jplg Y n IfjY n j4lg, nX1. Since each Y n is symmetric, it follows that fY ; Y n ðlÞ; nX1g are i.i.d. random variables. Clearly, (1) implies that X an PðjU n jX2bn Þo1. (2) lim sup gðÞ &0
n
Since n X
Y k IfjY k jplg ¼
k¼1
Un þ
n X
!, Y k ðlÞ
2,
k¼1
(2) shows that ! X n lim sup gðÞ an P Y IfjY k jplgX2bn k¼1 k &0 n X p2 lim sup gðÞ an PðjU n jX2bn Þo1. X
&0
ð3Þ
n
On the other hand, by Theorem A, we have ! X n X an P Y IfjY k jplgX2bn ¼ hðE½Y 2 IfjY jplg=4Þ. lim gðÞ k¼1 k &0 n
(4)
Combining (3) with (4), it follows there exists M40 such that hðE½Y 2 IfjY jplg=4ÞpMo1;
l40.
(5)
As limx!1 hðxÞ ¼ 1, on letting l ! 1 in (5) shows that EY 2 o1, and so EX 2 o1. We now show that (1) implies EX ¼ 0. Note that nEX ¼ S n ðS n nEX Þ, nX1. Hence, by (1) and Theorem A, we have X an PðjnEX jX2bn Þo1. (6) lim sup gðÞ &0
Since
P
n
n
an ¼ 1 and bn ¼ OðnÞ, (6) leads to EX ¼ 0. &
ARTICLE IN PRESS D. Li, A. Spa˘taru / Statistics & Probability Letters 76 (2006) 503–506
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2. Examples Example 1. Gut and Spa˘taru (2000b) proved that X 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P jSn jX n log log n ¼ s2 , lim 2 &0 n log n nX3 whenever EX 2 ¼ s2 o1 and EX ¼ 0. On account of Theorem 1, we may now assert conversely that X 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lim sup 2 P jS n jX n log log n o1 n log n &0 nX3 implies EX 2 o1 and EX ¼ 0. Example 2. Gut and Spa˘taru (2000a) proved also that if EX 2 ¼ s2 o1 and EX ¼ 0, then, for 0pdp1, lim 2dþ2 &0
X ðlog nÞd pffiffiffiffiffiffiffiffiffiffiffiffiffiffi mð2dþ2Þ 2dþ2 P jSn jX n log n ¼ s , n dþ1 nX1
where mð2dþ2Þ stands for the ð2d þ 2Þth absolute moment of the standard normal distribution. By Theorem 1, it follows that lim sup 2dþ2 &0
X ðlog nÞd pffiffiffiffiffiffiffiffiffiffiffiffiffiffi P jSn jX n log n o1 n nX1
entails EX 2 o1 and EX ¼ 0. Remark 2. Gut and Spa˘taru (2000b) pointed out that, E½X 2 ðlogþ logþ jX jÞZ o1 for some 0oZo1 and EðX Þ ¼ 0, X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 P jS n jX n log log n o1; 40. n log n nX3
under
the
weaker
assumption
that
(7)
It is easy to show that, under the same weaker assumption that E½X 2 ðlogþ logþ jX jÞZ o1 for some 0oZo1 and EðX Þ ¼ 0, for every 0pdo1, X ðlog nÞd pffiffiffiffiffiffiffiffiffiffiffiffiffiffi P jSn jX n log n o1; n nX1
40.
(8)
Thus one can construct i.i.d. random variables fX ; X n ; nX1g, such that EX 2 ¼ 1, EðX Þ ¼ 0, and (7) and (8) are fulfilled. This shows that Examples 1 and 2 are genuine converses to special cases of Theorem A. It is here the place to record two related results. Li et al. (1992) showed that, for bX2, bþ1=2 2 lim pffiffi ð 2Þ
& 2
X ðlog log nÞb pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P jS n jX n log log n ¼ 2b 2=pGðb þ 1=2Þ n nX3
if and only if E½X 2 ðlogþ logþ jX jÞb1 o1, EX 2 ¼ 1 and EX ¼ 0. It is also stated in Li et al. (1992) that, for b40, rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðlog nÞb pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2ðb þ 1Þ lim P jS jX n log log n ¼ n pffiffiffiffiffiffiffiffiffiffi b þ 1 n & 2ðbþ1Þ nX3 is equivalent to E½X 2 ðlogþ jX jÞb ðlogþ logþ jX jÞ1 o1, EX 2 ¼ 1 and EX ¼ 0.
ARTICLE IN PRESS 506
D. Li, A. Spa˘taru / Statistics & Probability Letters 76 (2006) 503–506
References Gut, A., Spa˘taru, A., 2000a. Precise asymptotics in the Baum–Katz and Davis law of large numbers. J. Math. Anal. Appl. 248, 233–246. Gut, A., Spa˘taru, A., 2000b. Precise asymptotics in the law of the iterated logarithm. Ann. Probab. 28, 1870–1883. Li, D., Wang, X., Rao, M.B., 1992. Some results on convergence rates for probabilities of moderate deviations for sums of random variables. Internat. J. Math. Math. Sci. 15, 481–498. Spa˘taru, A., 2004a. Exact asymptotics in loglog laws for random fields. J. Theoret. Probab. 17, 943–965. Spa˘taru, A., 2004b. Precise asymptotics for a series of T. L. Lai. Proc. Amer. Math. Soc. 132, 3387–3395.
