We study large partial sums, localized with respect to the sums of variances,
of a sequence of centered random variables. An application is given to the
distribution of prime factors of typical integers.Comment: 8 pages; small changes suggested by the referee. To appear in Stat.
  Prob. Letter

Ford, Kevin

Tenenbaum, Gerald

English

arXiv

À paraître dans Statistics and Probability Letters.We study large partial sums, localized with respect to the sums of variances, of a sequence of centered random variables. An application is given to the distribution of prime factors of typical integers

Tenenbaum, Gérald

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HAL Id: hal-00091337https://hal.archives-ouvertes.fr/hal-00091337v2Preprint submitted on 13 Mar 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Localized large sums of random variablesKevin Ford, Gérald TenenbaumTo cite this version:Kevin Ford, Gérald Tenenbaum. Localized large sums of random variables. 2007. ￿hal-00091337v2￿Localized large sums of random variablesKevin Ford 1,∗Department of Mathematics, University of Illinois at Urbana-Champaign,1409 West Green St., Urbana, IL, 61801, USAGérald TenenbaumInstitut Élie Cartan, Université Henri–Poincaré Nancy 1,B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, FranceAbstractWe study large partial sums, localized with respect to the sums of variances, of a sequenceof centered random variables. An application is given to the distribution of prime factors oftypical integers.Key words: sums of random variables, prime divisors1991 MSC: 60G50, 11N25Dedicated to the memory of Walter Philipp1 IntroductionConsider random variables X1, X2, . . . with EXj = 0 and EX2j = σ2j . LetSn = X1 + · · · +Xn, s2n = σ21 + · · · + σ2n,and assume that (a) sn → ∞ as n→ ∞.∗ Corresponding author.Email addresses: ford@math.uiuc.edu (Kevin Ford),gerald.tenenbaum@iecn.u-nancy.fr (Gérald Tenenbaum).1 Research supported by National Science Foundation Grants DMS-0301083 and DMS-0555367Preprint submitted to Elsevier 13 March 2007Given a positive function fN  1 + 1/N , we are interested in the behavior ofI = lim infN→∞maxN<s2nNfN|Sn|/sn.If we replace lim inf by lim sup, it immediately follows from the law of the iteratedlogarithm that I = ∞ almost surely when fN is bounded. Our results answer aquestion originally raised, in oral form, by A. Sárközy and for which a partial an-swer had previously been given by the second author, see Chap. 3 of Oon (2005).2 Independent random variablesAssume that the Xj are independent. Then ES2n = s2n. In addition to condition (a),we will work with two other mild assumptions, (b) sj+1/sj  1 when sj > 0and (c) for every λ > 0, there is a constant cλ > 0 such that if n is large enoughand s2m > 2s2n, thenP (|Sm − Sn|  λsm)  cλ.Condition (b) says that no term in Sn dominates the others. Condition (c) followsif the Central Limit Theorem (CLT) holds for the sequence of Sn, since CLT for Snimplies CLT for Sm−Sn as (m−n) → ∞. For example, (c) holds for i.i.d. randomvariables, under the Lindeberg condition∀ε > 0, limn→∞∑1jnE(X2j /s2n : |Xj| > εsn)= 0and the stronger Lyapunov condition∃δ > 0 :∑1jnE|Xj|2+δ = o(s2+δn ).Condition (c) is weaker, however, than CLT.Theorem 1 (i) Suppose (a), (b), and fN = (logN)M for some constant M > 0.Then I <∞ almost surely.(ii) Suppose (a), (b), (c) and fN = (logN)ξ(N) with ξ(N) tending monotonicallyto ∞. Then I = ∞ almost surely.Remark. In the first statement of the theorem we show in fact that almost surelyI  15√M + 1(maxsj>0 sj+1/sj)2.Lemma 2 (Kolmogorov’s inequality, 1929) We haveP(max1jk |Sj|  λsk)  1/λ2 (k  1).2Proof of Theorem 1. By (a) and (b), there is a constant D so that sj+1/sj  D forall large j. Defineh(n) := max{k : s2k  n} (n ∈ N∗),so that the conditions N < s2n  NfN and h(N) < n  h(NfN) are equivalent.We first consider the case when fN := (logN)M . LetNj := j(M+3)j, t(j) := (M + 1)(log j)/ log 2, Hj := 2t(j),andUj := h(Nj), Uj,t := h(2tNj) (0  t  t(j)), Vj := h(HjNj) = Uj,t(j).It is possible that Uj,t+1 = Uj,t for some t. Note that for large j, HjNj  NjfNj .Let k be a constant depending only on M and D. For j  1 define the eventsAj := {|SVj |  sUj+1},Bj :=⋂0tt(j)−1Bj,t where Bj,t :={maxUj+1,tnUj+1,t+1|SUj+1,t+1 − Sn|  ksUj+1,t},Cj := {|SUj+1 − SVj |  2sUj+1}.By (b) and the definition of h(N), we haveD−1√2tNj  sUj,t √2tNj (1)for all j, t. It follows from Lemma 2 thatP(Aj)  D2HjNjNj+1 D2j2.Thus,∑j1 P(Aj) <∞ and hence almost surely there is a j0 so that Aj occurs forj  j0. Applying Lemma 2 again yieldsP(Bj,t) s2Uj+1,t+1 − s2Uj+1,tk2s2Uj+1,t D22t+1Nj+1k22tNj+1=2D2k2.If k = 3D√M + 1, thenP(Bj) (1 − 2D2k2)t(j) 1j1/2for large j. Also by Lemma 2, P(Cj)  34 , and since Bj and Cj are independent,∑j1P(BjCj) = ∞.3Since the events BjCj are independent, the Borel–Cantelli lemma implies that al-most surely the events BjCj occur infinitely often. Thus, the event AjBjCj occursfor an infinite sequence of integers j. Take such a index j, let n ∈ [Uj+1, Vj+1] andUj+1,g−1 < n  Uj+1,g, where 1  g  t(j + 1). We have by several applicationsof (1)|Sn|  |SVj | + |SUj+1 − SVj | +∑0tg−2|SUj+1,t − SUj+1,t+1 | + |Sn − SUj+1,g−1 | 3sUj+1 + k∑0tg−1sUj+1,t{3 + k(1 + 21/2 + · · · + 2(g−1)/2)}√Nj+1 5k√2g−1Nj+1 5kDsn = 15D2(M + 1)1/2sn.This completes the proof of part (i) of the theorem, sinceVj+1  h(12jM+1Nj)  h(Nj logM Nj)for large j.Now suppose fN = (logN)ξ(N) with ξ(N) tending monotonically to ∞.Let λ > 0 be arbitrary and defineK := 2D2. LetN∗1 be so large that fN∗1  K. Forj  1 let N∗j+1 = N∗jKu(j), where u(j) := log fN∗j / logK. PutU∗j := h(N∗j ), U∗j,t := h(KtN∗j ) (0  t  u(j)).Let Jj := [U∗j , U∗j+1] andYj := maxn∈Jj|Sn|/sn.We haveu(j)  1 ⇒ N∗j+1  KN∗j ⇒ u(j)/ log j → ∞.Therefore, by (c), if j is sufficiently large thenP(Yj  λ/2) ∏1tu(j)P(|SU∗j,t − SU∗j,t−1| 12λ(sU∗j,t + sU∗j,t−1))∏1tu(j)P(|SU∗j,t − SU∗j,t−1|  λ√KtN∗j) (1 − cλ)u(j) 1j2.Thus ∑j1P(Yj  λ/2) <∞.Almost surely, Yk  λ/2 for only finitely many k.4Theorem 1 has an analog for Brownian motion, which follows from Theorem 1 andthe invariance principle.Theorem 3 Let W (t) be Brownian motion on [0,∞). If fN = (logN)M with fixedM > 0, then almost surelyI = lim infN→∞maxN<tNfN|W (t)|√t<∞.If fN = (logN)ξ(N) with ξ(N) → ∞, then I = ∞ almost surely.Theorem 3 can be proved directly and more swiftly using the methods used toestablish Theorem 1. By invariance principles (e.g. Philipp , 1986), one may deducefrom Theorem 3 a version of Theorem 1 where stronger hypotheses on the Xj areassumed. As it stands, now, however, Theorem 1 does not follow from Theorem 3.3 Dependent random variablesThe conclusions of Theorem 1 can also be shown to hold for certain sequences ofweakly dependent random variables by making use of almost sure invariance prin-ciples. We assume that (d) there exists a sequence of i.i.d. normal random variablesYj with EY 2j = σ2j , defined on the same probability space as the sequence of Xj ,and such that if Zn = Y1 + · · · + Yn, then|Sn − Zn| = O(sn) a.s.Of course the variables Yj are dependent on theXj , but not on each other. Property(d) has been proved for martingale difference sequences, sequences satisfying cer-tain mixing conditions, and lacunary sequencesXj = {njω} with inf nj+1/nj > 1,ω uniformly distributed in [0, 1] and {x} is the fractional part of x. See e.g. Philipp(1986) for a survey of such results.Theorem 4 (i) Suppose (a), (b), and (d). If fN := (logN)M for some constantM > 0, then I <∞ almost surely.(ii) Let ξ(N) tend monotonically to ∞ and set fN := (logN)ξ(N). Then I = ∞almost surely.By (d),I = O(1) + lim infN→∞maxN<s2nNfN|Zn|/sn,and we apply Theorem 1 to the sequence of Yj . The variable Zn is normal withvariance s2n, hence (c) holds.54 Prime factors of typical integersConsider a sequence of independent random variables Yp, indexed by prime num-bers p, such that P(Yp = 1) = 1/p and P(Yp = 0) = 1 − 1/p. We can think of Ypas modelling whether or not a “random” integer is divisible by p. As EYp = 1/p,we form the centered r.v.’sXp = Yp − 1/p (we may also define Xj for non-prime jto be zero with probability 1). LetTn =∑pnYp, Sn =∑pnXp.We have EX2p = (1 − 1/p)/p, hence by Mertens’ estimates2n =∑pn1p− 1p2= log2 n+O(1).Here and in the sequel, logk denotes, for integer k  2, the k-fold iterated log-arithm. Since E|Xp|3  1/p, the Lyapunov condition holds with δ = 1. Then(a), (b) and (c) hold, and therefore the conclusion of Theorem 1 holds. Here takeD = maxn2 sn+1/sn since s1 = 0.Let ω(m, t) denote the number of distinct prime factors of m which are  t. Thesequence {Tn : n  1} mimics well the behavior of the function ω(m,n) for a“random” m, at least when n is not too close to m. This is known as the Kubiliusmodel. It can be made very precise, see (Elliott , 1979, Ch. 3, especially pp. 119–122) and Tenenbaum (1999) for the sharpest estimate known to date. Suppose ris an integer with 2  r  x and r = x1/u, ωr(m) = (ω(m, 1), . . . , ω(m, r)) andsuppose Q is any subset of Zr. Then, given arbitrary c < 1, and uniformly in x, rand Q, we have1x|{m  x : ωr(m) ∈ Q}| = P((T1, . . . , Tr) ∈ Q)+O(x−c + e−u log u). (2)An analog of Theorem 1, established by parallel estimates, provides via (2) infor-mation about localized large values of/(m, t) := |ω(m, t) − log2 t|/√log2 t.Theorem 5 (i) LetM > 0 be fixed, fN := (logN)M and putK := 30D2√M + 1.If g = g(m) → ∞ monotonically as m → ∞ in such a way that g2fg2  log2mfor large m, then for a set of integers m of natural density 1, 2 we haveming(m)Ng(m)2maxN<log2 tNfN/(m, t)  K.2 A subset A of N∗ is said to have natural density 1 if |A ∩ [1, x]| = x+ o(x) as x → ∞.6(ii) Let ξ(N) → ∞ in such a way that fN := (logN)ξ(N)  N . Suppose thatg(m) → ∞ monotonically as m→ ∞, that g(m)  (log2m)1/10, and letIm := ming(m)NNfN log2 mmaxNlog2 tNfN/(m, t).Then, Im → ∞ on a set of integers m of natural density 1.We follow the proof of Theorem 1. Keeping the notation introduced there, we seethat for large J ,P ⋂Jj3J/2AjBjCj  ∑Jj3J/2D2j2+∏Jj3J/2(1 − 34√j) 1J·For large G, define J by NJ+1 < G  NJ+2. Then G5/3 > N3J/2+2 and J M(logG)/ log2G. Thus, for large G,P(minGNG5/3maxh(N)<nh(NfN )|Sn|sn K) 1 −O(1J) 1 −O(log2GlogG).The direct number theoretic analog of |Sn|/sn is/̃(m, t) :=∣∣∣ω(m, t) − ∑pt 1/p∣∣∣√∑pt(1 − 1/p)/p.By (2), if G is large and G √log2 x (so that G5/3fG5/3  (log2 x)7/8), then1x∣∣∣∣{m  x : minGNG5/3maxh(N)<nh(NfN )/̃(m, t)  K}∣∣∣∣  1 −O(log2GlogG).Since /̃(m, t) = /(m, t) +O(1/√log2 t), the first part of the theorem follows.The second part is similar. Note that ω(n, x) − ω(n, x1/√log2 x)√log2 x forn  x, and, for brevity, write g = g(√x). By (2) with u :=√log2 x, we have, forany fixed K and large x,1x∣∣∣∣∣{m  x : minNgNfN log2 mmaxN<log2 tNfN/̃(m, t)  K}∣∣∣∣∣ 1x∣∣∣∣∣∣∣{√x  m  x : minNgNfN L (x)maxNlog2 tNfN/̃(t)  K + 2}∣∣∣∣∣∣∣ +1√x P infNgNfN L (x)maxh(N)<nh(NfN )|Sn|sn K + 2 +O( 1log2 x),7where L (x) := log2 x − 12 log3 x. Since fN  N , we have N∗j+1  (N∗j )2 in thenotation of the proof of Theorem 1. The interval[(log2 x)1/10,L (x)1/2]therefore contains at least one interval Jj . By the proof of Theorem 1, for large x,the probability above does not exceed∑jj0 1/j2  1/(j0 − 1), where j0 → ∞ asx→ ∞.Remarks. The upper bound g2 of N in the first part can be sharpened. By thesame methods, similar results can be proved for a wide class of additive arithmeticfunctions r(m, t) =∑pa‖m r(pa) in place of ω(m, t).Acknowledgment. The authors are indebted to Walter Philipp for helpful discus-sions on the use of almost sure invariance principles.ReferencesP.D.T.A. Elliott, 1979. Probabilistic Number Theory, I, Springer-Verlag, New York.S.M. Oon, 2005. Construction des suites binaires pseudo-aléatoires, Thèsed’université, Nancy, Université Henri Poincaré–Nancy 1, UFR STMIA , 110 pp.W. Philipp, 1986. Invariance principles for independent and weakly dependent ran-dom variables, in Dependence in Probability and Statistics (Oberwolfach, 1985),225–268, Progr. Probab. Statist. 11, Birkhäuser Boston, Boston, MA.G. Tenenbaum, 1999. Crible d’Ératosthène et modèle de Kubilius, in: K. Győry, H.Iwaniec, J. Urbanowicz (eds.), Number Theory in Progress, Proceedings of theconference in honor of Andrzej Schinzel, Zakopane, Poland 1997, 1099–1129,Walter de Gruyter, Berlin, New York.8

Localized large sums of random variables

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Localized large sums of random variables

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