On the asymptotic growth rate of the sum of p-adic-valued independent identically distributed random variables

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1 Arab. J. Math : DOI /s Arabia Joural of Mathematics Kumi Yasuda O the asymtotic growth rate of the sum of -adic-valued ideedet idetically distributed radom variables Received: 7 December 013 / Acceted: 5 May 014 / Published olie: 19 Jue 014 The Authors 014. This article is ublished with oe access at Srigerlik.com Abstract The -adic semistable laws are characterized as weak limits of scaled summads of -adic-valued, rotatio-symmetric, ideedet, ad idetically distributed radom variables whose tail robability satisfies some coditio. I this article it is verified such scaled sums do ot coverge i robability, ad some more recise estimates, corresodig to the law of iterated logarithm i the real-valued settig, are give to the asymtotic growth rate of the sum. The critical scalig order is exlicitly give, over which the scaled sum almost surely coverges to 0. O the other had, uder the critical order, the limit suerior of the -adic orm of the scaled sum diverges almost surely. Furthermore it is show that, at the critical order, a crucial chage of the asymtotic behavior of the scaled sum occurs accordig to the decay of the tail robability of the radom variables. I this situatio, the critical value for the order of the tail robability is also foud. Mathematics Subject Classificatio 60G50 11F85 60F05 1 Itroductio ad mai theorems Limit theorems for sums of real-valued ideedet radom variables are oe of the core subjects i robability theory. It is kow that weak limits of sequeces of ormalized sums of ideedet ad idetically distributed i.i.d. radom variables form the whole set of stable distributios [1,3]. The cetral limit theorem assures that, uder some additioal coditios, the limits are the stadard ormal distributio. However, they are ot coverget i stroger seses, such as almost surely or i robability, ad i this situatio the law of the iterated logarithm gives a more recise estimates for the growth rate of the sums [,3]. Cocerig with the -adic-valued radom variables, limit distributios of the scaled sums of rotatiosymmetric i.i.d. ξ i i 1,,... are discussed i [4]. Here, a rotatio-symmetric radom variable is such ξ K. Yasuda B Faculty of Busiess ad Commerce, Keio Uiversity, Hiyoshi, Kohoku-ku, Yokohama 3-851, Jaa kumi@a6.keio.j

2 450 Arab. J. Math : that Pξ X Pξ ux for ay measurable subset X of the -adic field ad ay uit u i the rig of the -adic itegers. I case the ξ i are bouded ad o-degeerate, the ultra-metric roerty of the -adic orm brigs a differet situatio from the Euclidea case; the distributio of the sum, without ay scalig, coverges to the ormalized Haar measure o the miimum disc cetered at the origi that cotais the suort of ξ i.o the other had, if we cosider ubouded ξ i, -adic semistable laws aear as the weak limit distributios of the scaled sums. I this article, we will show the covergece to the -adic semistable laws does ot follow i stroger seses, ad determie the recise growth rate of the sum. Throughout this article, we fix a rime iteger. For a o-zero iteger z,the-adic orm of z is defied by z m,wherem is the maximum iteger such that m divides z.the-adic orm is aturally exteded to ratioal umbers, if we defie 0 0ad z z z z for itegers z ad z 0. The field of -adic umbers, deoted by Q, is the toological closure of Q with resect to the -adic orm. The o-zero elemets of Q are idetified with formal series x im a i i with m Z, a i {0, 1,,..., 1}, a m 0,adtheyare ormed by x m.the-adic orm satisfies the ultra-metric iequality : x + x max { x, x }, ad this characteristic roerty is frequetly used for some estimatios i the followigs. A character o Q is a cotiuous homomorhism o the additive grou of -adic umbers to the multilicative grou of comlex umbers with absolute value 1. For -adic umbers x, let us defie { 1, if x 0orx im a i i with m 0, ϕ 1 x : ex π 1 1 im a i i, if x im a i i with m 1. The ϕ 1 is a character o Q, ad ay character ϕ is of the form ϕx ϕ y x : ϕ 1 yx for some y Q.For a distributio μ o the field of -adic umbers, its characteristic fuctio is defied o the grou of characters by ˆμϕ : Q ϕxμdx. The characteristic fuctio is cosidered to be a fuctio o Q through the idetificatio ϕ y y, ad writte as ˆμy : ˆμϕ y, y Q. Defiitio 1.1 For α > 0, a distributio μ o the field of -adic umbers is called a -adic α-semistable law, if its characteristic fuctio is give by for some costat c > 0. ˆμy e c y α, y Q, For -adic-valued i.i.d. ξ i i 1,,..., we shall deote their tail robability by T s : P ξ i s, s 0, ad let S : i1 ξ i be their artial sums. The -adic semistable laws are characterized as limit distributios of scaled sums of rotatio-symmetric i.i.d. as follows. Proositio 1. [4] Theorem A distributio μ o the field of -adic umbers is semistable, if ad oly if it is a o-degeerate limit distributio of the scaled sum S k for some rotatio-symmetric i.i.d. ξ i i 1,,... ad some icreasig sequece {k} 1,,... of atural umbers satisfyig su kt +l < + for ay iteger l. I what follows, the iteger art of a real umber t is deoted by [t]. For a radom variable X, LX deotes the law of X, ad ˆLX its characteristic fuctio. We let μ α be the α-semistable law with the characteristic fuctio ˆμ α y e y α. Proositio 1.3 [4] Proositio 4 Suose the tail robability T of the rotatio symmetric i.i.d. ξ i is give by T α L, Z, L+1 for some α > 0 ad some sequece of ositive umbers L satisfyig lim L 1. If we ut N : C α T 1,C α : α 1 α+1 1, the the law of the scaled sum S [N] coverges to μ α as. Uder the coditio of Proositio 1.3, we will determie the growth rate of S [N].

3 Arab. J. Math : Theorem 1.4 Let β>0, ad ut c [βlog ]. i If β> α log 1,thelim su +c S [N] 0, ad +c S [N] coverges to 0 almost surely. ii If β< α log 1,thelim su +c S [N] + almost surely. We shall examie ext what haes at the critical order c log α log. As the followig theorem shows, the both cases lim su +c S [N] 0ad+ may occur accordig to the order of the covergece L+1 L 1. Theorem 1.5 Let c [ ] log α log, ad suose L+1 L log γ/log for with some γ>0. i If γ>αlog, the lim su +c S [N] 0, ad +c S [N] coverges to 0 almost surely. ii If γ α log, the lim su +c S [N] + almost surely. Proofs to Theorems 1.4 ad 1.5 will be give i Sect. 4. Covergece of the scaled sum L+1 I the followigs let α>0, {L} be a sequece of ositive umbers such that lim L 1, ad ξ i be -adic-valued, rotatio symmetric i.i.d. with the tail robability T α L. Besides the weak covergece cofirmed i Proositio 1.3, let us verify whether the scaled sum S [N] coverges i ay stroger sese. Proositio.1 Uder the coditio of Proositio 1.3, the scaled sum S [N] does ot coverge almost surely. Proof We have [N] S [N] S [N] ξ i + i1 [N] i[n]+1 [N] i[n]+1 ξ i [N] i1 ξ i 1 S [N], 1 ad by Proositio 1.3 ad 1 1iQ,thelawof1 S [N] coverges to μ α. Sice the radom variables [N] i[n]+1 ξ i ad 1 S [N] are ideedet, comarig the characteristic fuctios of the both sides of 1gives ˆL S [N] S [N] y ˆL [N] i[n]+1 ξ i ˆL 1 S [N] y y ˆL 1 S [N] y μˆ α y e y α,, for each -adic umber y. If we suose S [N] is coverget almost surely, the S [N] S [N] coverges to 0 almost surely. The the left-had side of should ted to 1, which is a cotradictio. Furthermore, we ca show the covergece i Proositio 1.3 is ot eve i robability. Proositio. Uder the coditio of Proositio 1.3, the scaled sum S [N] does ot coverge i robability. ξ i

4 45 Arab. J. Math : Proof Take a radom variable X α whose law is μ α. If we suose S [N] coverges to X α i robability, the for ay ε>0wehavep S [N] X α >ε 0as. The ultra-metric roerty imlies S [N] { S [N] max S [N] X α, } S [N] X α, ad therefore P S [N] S [N] >ε P S [N] X α >ε + P S [N] X α >ε 0. This shows that S [N] S [N] coverges to 0 i robability, ad hece i law. The we have ˆL S [N] S [N] y 1forayy Q, which cotradicts agai. 3 Limit suerior of the orm of the scaled sum Next let us study the limit suerior of S [N]. For a iteger m ad atural umbers k > l, ut A m k, l : P S k S l < m. Lemma 3.1 A m k, l 1 1 T m k l 1 +. Proof Sice S k S l S k 1 S l + ξ k, the ultra-metric roerty imlies that, S k S l < m holds oly if either S k 1 S l, ξ k < m or S k 1 S l, ξ k m. Hece we ca see A m k, l P S k 1 S l < m, ξ k < m + P S k 1 S l m, ξ k m A m k 1, l 1 T m + 1 A m k 1, lt m A m k 1, l 1 T m + T m, sice the radom variables S k 1 S l ad ξ k are ideedet. Iductively we ca derive that A m k, l A m l + 1, l 1 T m k l 1 + T m k l 1 T m j 1 T m 1 T m k l 1 + T m 1 1 T m k l T m 1 1 T m k l 1 +. j0 Proositio 3. Uder the coditio of Proositio 1.3, lim su S [N] +, a.s. Proof Sice T 0ad L L 1 1, we ca see N N 1 C α T 1 1 α L +, L 1 as. The we ca assume [N] > [N 1] for large. For atural umbers ad s, ut R,s : P S [N] S [N 1] +s, the accordig to Lemma 3.1 it follows that R,s 1 A +s [N], [N 1] T +s [N] [N 1]. 3

5 Arab. J. Math : Here let us ut N [N] [N 1] ad t T +s,the L Nt N N 1 1t C α α L 1 1 α L L 1 αs L + s C α L αs L + 1 C α L 1 α L L 1 L + L + 1 L + s L + s 1 Sice L 1 1, we ca take M 1suchthat α/ < above imlies α+s L + s t t t 1 N + 1 C α αs α/ s 1 α α/ 1 N + 1 C α 3αs/ 1 α/, for M. Puttig η : C α 3αs/ 1 α/,3 reads R,s α L t. L 1 L L 1 < α/ holds for all M. The the 1 η N. 4 N + 1 Sice N N N as, the right-had side of 4 teds to 1 1 e η > 0. Hece we obtai 1 R,s +. Sice the evets S [N] S [N 1] +s are ideedet for 1,,..., Borel s theorem yields P S [N] S [N 1] +s for ifiitely may 1. 5 Let us show 5 imlies P S [N] s for ifiitely may 1. Suose S [N] ω < +s for all large, the by the ultra-metric roerty S [N] ω S [N 1] ω { max S [N] ω, S [N 1] ω } < +s holds for all large. Whereas such a evet occurs with robability 0 accordig to 5, ad thus we obtai P S [N] s for ifiitely may 1. Cosequetly lim su S [N] s holds with robability 1, ad lettig s,weobtailimsu S [N] + almost surely. I what follows we cosider a o-decreasig sequece of o-egative itegers {c }, ad look ito the behavior of lim su +c S [N]. We will see it may coverge or diverge accordig to the sequeces {c } ad {L}. PutS : 1 αc L+c L. Lemma 3.3 i If S < +, thelim su +c S [N] 0, ad +c S [N] coverges to 0 almost surely. ii If S +,thelim su +c S [N] + almost surely. Proof I case c is bouded, we have S + ad lim su +c S [N] +. Ideed, we ca take M N such that L+1 L > 1 L+1 for ay M, by the coditio L 1. If we ut C : su c,thewe have L + c L + 1 L + L L L + 1 L + c 1 c 1 C L + c 1 >,

6 454 Arab. J. Math : ad αc αc for M, which yields S + clearly. O the other had, Proositio 3. imlies lim su +c S [N] C lim su S [N] +, ad thus the case where c is bouded corresods to the assertio ii. We shall assume c + i the followigs. i Fix a atural umber s, ad defie P,s : P max 1 i [N] S i +c s for 1,,...By the ultra-metric roerty, max 1 i [N] S i +c s holds if ad oly if max 1 i [N] ξ i +c s, ad sice ξ i are ideedet, we have P,s 1 P ξ i < +c s, i 1,,...,[N] [N] 1 i1 Here ote that for t > 0 ad a atural umber N, P ξ i < +c s 1 1 T +c s [N] t N N k1 N k t k Nt N 1 j1 Nt N j1 Nt N j1 N j t j N j+1 t j+1, if N is odd, N j t j N j+1 t j+1 t N N j t j N j+1 t j+1, if N is eve, 7 ad the each term i these sums is estimated as N N t j t j+1 j j + 1 N! j + 1!N j! t j j + 1 N jt N! j + 1!N j! t j 3 Nt. 8 By the assumtios L+1 L 1adc +, there exists M 1suchthat α/ < L+1 L c > s for all M. Let M ad ut N [N], t T +c s,the < α/ ad Nt Nt C α αc s L + c s L C α αc L + 1 L + s L L + 1 L + c s L + c s 1 < C α αc s α/ c s C α αc s/. 9

7 Arab. J. Math : If we take M M sufficietly large, the for ay M the last term i 9 is less tha 3, ad thus the right-had side of 8 is ositive. Hece we ca derive from 6ad7that P,s Nt C α αc s L + c s L C α αc s L + c L + c 1 L L + c < C α αc s L + c α/ s L C α 3αs/ αc L + c, L rovided M. Accordigly, the assumtio S < + imlies M P,s C α 3αs/ S < +, L + c L + c 1 L + c s L + c s + 1 ad by Borel Catelli s lemma, P max S i +c s for ifiitely may 0. 1 i [N] Hece lim su +c S [N] lim su c S [N] < s holds almost surely, ad sice s is arbitrary, we coclude that lim su +c S [N] 0. The it is clear that +c S [N] coverges to 0 almost surely. ii Sice N N 1 C α T 1 1 α L + L 1 as, we ca assume [N] > [N 1] for large. Let us ut Q,s : P S [N] S [N 1] +c +s for atural umbers ad s, the Lemma 3.1 yields Here for 0 < t < 1 ad a atural umber N,wesee 1 1 t N N k1 N k Q,s 1 A +c +s[n], [N 1] T +c +s [N] [N 1]. 10 t k Nt NN 1 t + N N j j 1 t j 1 N j t j, if N is eve, Nt NN 1 t + N 1 N j j 1 t j 1 N j t j + t N Nt NN 1 t + N 1 N j j 1 t j 1 N j t j, if N is odd,

8 456 Arab. J. Math : ad the each term i these sums is estimated as N t j 1 j 1 Sice L+1 L N j t j j 1 t N! j 1!N j + 1! N! j 1!N j + 1! t j 1 1 Nt N j + 1 t j 1adT 0as, there exists M 1 such that α/ < L+1 L < α/ ad T +c +s < 1 hold for all M.PutN [N] [N 1] ad t T +c +s, the for M we have Nt N N 1 + 1T +c +s C α 1 α L αc +s L + c + s + T +c +s L 1 L C α 1 α L αc L + 1 +s L 1 L +T +c +s < C α 1 α L L 1 C α 1 α L L 1 L + L + 1 L + c + s L + c + s 1 αc +s α/ c +s + T +c +s αc+s/ + T +c +s. 1 L Sice L 1 1, αc+s/ 0, ad T +c +s 0as,wecatakeM > M such that the right-had side of 1 islesstha1foray M. The for M the right-had side of 11 is ositive, ad hece 1 1 t N Nt NN 1 t.ifm is sufficietly large, we ca assume N N 1, adthe10 leads to Q,s 1 NN 1 Nt t > 1 Nt 1 N t > 1 Nt 1 1 [N] [N 1]T +c +s 1 4 N N 1T +c +s 1 4 C α 1 α L αc +s L + c + s L 1 L 1 4 C α 1 α L αc +s L + c L 1 L L + c + 1 L + c + L + c L + c + 1 L + c + s L + c + s 1 > 1 4 C α 1 α α/ αc +s L + c α/ s L 1 4 C α 1 α/ 3αs/ αc L + c. L

9 Arab. J. Math : The the assumtio S + imlies Q,s 1 4 C α 1 α/ 3αs/ L + c αc +, L M M ad sice the evets S [N] S [N 1] +c +s are ideedet for 1,,..., Borel s theorem shows P S [N] S [N 1] +c +s for ifiitely may If we suose S[N] ω < +c+s for all large, the by the ultra-metric roerty, S[N] ω S [N 1] ω { S[N] max ω, S[N 1] ω } < max { +c+s, 1+c 1+s } +c +s holds for all large. Whereas the robability of such a evet is 0 accordig to 13, ad cosequetly we have lim su +c S [N] c S [N] s with robability 1. Lettig s, we obtai lim su +c S [N] + almost surely. Followig Lemma 3.3, we shall look for the critical order of the sequece {c } over which the scaled sum +c S [N] coverges to 0 almost surely. For the first try, let us examie ositive owers of. Proositio 3.4 For ay β>0 ad ε>0,c [β ε ] gives +c S [N] 0 almost surely. Proof Accordig to the assumtio L+1 L 1, take M 1suchthat L+1 L < α/ holds for ay M. 1/ε For each atural umber l, c [β ε ] takes the value l if ad oly if lβ < l+1 1/ε, β ad therefore M αc L + c L lc M lβ 1/ε < 1/ε l+1 β l + 1 1/ε β lc M αl L + 1 L l 1/ε αl α/ l β β 1/ε l + 1 1/ε l 1/ε αl/ β 1/ε lc M lc M l + 1 1/ε αl/. Here uttig d l : l + 1 1/ε αl/,wehave d l+1 d l l + 1/ε α/ α/ < 1 l + 1 L + L + 1 L + l L + l 1 as l. The by d Alembert s ratio test the sum M αc L+c L, ad hece S is coverget. Thus the assertio follows by Lemma 3.3. This roositio shows that ay ositive ower of is too fast for a cadidate of the critical order of c.the correct order is c β log for some β>0, as is give i Theorem 1.4.

10 458 Arab. J. Math : Proofs of Theorems Proof Theorem 1.4 iputαβ log 1 + δ, δ>0, the αc < αβ log 1 α 1 δ.bythe assumtio L+1 L 1, we ca take M 1 such that L+1 L < αδ/1+δ holds for all M. The for M it follows that ad therefore L + c L < M L + 1 L + L L + 1 L + c L + c 1 αδ/1+δ c αδ/1+δ β log δ/, αc L + c L α M 1 δ/ < +. This imlies S < +, ad the the assertios follow by Lemma 3.3. ii Put αβ log 1 δ, δ>0, the αc αβ log 1+δ.ForsomeM 1, we ca suose > αδ/1 δ for all M,adthe L+1 L L + c L > L + 1 L + L L + 1 L + c L + c 1 αδ/1 δ c αδ/1 δ β log δ/. Hecewehave S M αc L + c L 1+δ/ +, M ad Lemma 3.3 imlies lim su +c S [N] + a.s. Our ext iterest is i what haes at the critical order c where L is costat for 1. ] Examle 4.1 I case c 1for 1, we have Ideed, we see [ log α log 1 ad L+1 L lim su S αc +c S [N] +. 1 log α α log log α log. Let us examie the simlest case 1 +. It caot be cocluded that the sum with the secific {c } i the examle ] diverges for all sequeces {L}. As Theorem 1.5 shows, at the critical order c, the covergece/divergece of 1 [ log α log lim su +c S [N] deeds o the order of the covergece L+1 L 1.

11 Arab. J. Math : Proof Theorem 1.5Letussee L + c L +c 1 k +c 1 k Lk + 1 Lk γ log log k/ log k e e γ +c 1 k log log k/ log k, ad estimate H : +c 1 k log log k log k. i Fix umbers ρ ad ρ such that α log γ The for M, wehave1 ρ log α log 1adthe Sice c log α log 1 <ρ <ρ<1, ad take a large iteger M so that log M α log 1 ρ. log log 1 ρ α log α log ρ log α log. log log k log k log log+c log+c holds for k, + 1,..., + c 1, it follows that H c log log + c log + c ρ log α log log log + c. log + c log log t Cosider a fuctio f t : log t for t > e e,the f t >0 ad its derivative f log log t 1 t is egative tlog t ad icreasig for sufficietly large t. Therefore if is sufficietly large, we have or equivaletly f + c f + f c log log 1 log f log α log f 1 α log log log 1, log f + c f 1 1 log log 1 α log log 1 1 α log 1 f log log 1 log log. 14 Sice the right-had side of 14 teds to 1 as,wecatakem M such that f +c /f α log γρ for all M,adthe Put θ : ρ ρ > 1, the the above leads us to H ρ log α log α log ρ log log γρ f γρ. L + c L e γ ρ log log /γρ e θ log log 1 log θ

12 460 Arab. J. Math : for M, ad cosequetly L + c αc αlog /α log 1 1 L log θ M mm α α log θ 1 M M 1 1 dt < +, tlog t θ sice θ>1. This imlies S < +, ad our assertio follows by Lemma 3.3. log log k log log ii Sice log k log for k, + 1,..., + c 1, The which leads to H c log log log L + c L log log log log log α log log α log. e γ log log /α log e log log 1 log, αc L + c L αlog /α log 1 log 1 log 1 t log t dt +. Thus we obtai S +, ad Lemma 3.3 gives lim su +c S [N] +. Ackowledgmets This work was suorted by Grat-i-Aid for Scietific Research B Oe Access This article is distributed uder the terms of the Creative Commos Attributio Licese which ermits ay use, distributio, ad reroductio i ay medium, rovided the origial authors ad the source are credited. Refereces 1. Gedeko, B.V.; Kolmogorov, A.N.: Limit Distributios for Sums of Ideedet Radom Variables. Addiso-Wesley, Lodo Ito, K.: A Itroductio to Probability Theory. Cambridge Uiversity Press, Cambridge Petrov, V.V.: Limit Theorems of Probability Theory. Oxford Sciece Publicatios, Oxford Yasuda, K.: O ifiitely divisible distributios o locally comact Abelia grous. J. Theor. Probab. 13,