We prove Kolmogorov's law of the iterated logarithm for noncommutative
martingales. The commutative case was due to Stout. The key ingredient is an
exponential inequality proved recently by Junge and the author.Comment: Revise

Zeng, Qiang

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www.imstat.org/aihpAnnales de l’Institut Henri Poincaré - Probabilités et Statistiques2015, Vol. 51, No. 3, 1124–1130DOI: 10.1214/14-AIHP603© Association des Publications de l’Institut Henri Poincaré, 2015Kolmogorov’s law of the iterated logarithm for noncommutativemartingalesQiang ZengDepartment of Mathematics, University of Illinois, Urbana, IL 61801, USA. E-mail: zeng8@illinois.eduReceived 5 August 2013; revised 12 January 2014; accepted 22 January 2014Abstract. We prove Kolmogorov’s law of the iterated logarithm for noncommutative martingales. The commutative case was dueto Stout. The key ingredient is an exponential inequality proved recently by Junge and the author.Résumé. Nous prouvons la loi de Kolmogorov du logarithme itéré pour des martingales non-commutatives. Le cas commutatif aété établi par Stout. L’ingrédient clé est une inégalité exponentielle prouvée récemment par Junge et l’auteur.MSC: 46L53; 60F15Keywords: Law of the iterated logarithm; Noncommutative martingales; Quantum martingales; Exponential inequality1. IntroductionIn probability theory, law of the iterated logarithm (LIL) is among the most important limit theorems and has beenstudied extensively in different contexts. The early contributions in this direction for independent increments weremade by Khintchine, Kolmogorov, Hartman–Wintner, etc.; see [1] for more history of this subject. Stout generalizedKolmogorov and Hartman–Wintner’s results to the martingale setting in [15,16]. The extension of LIL for independentsums in Banach spaces were due to Kuelbs, Ledoux, Talagrand, Pisier, etc.; see [12] and the references therein formore details in this direction. In the last decade, there has been new development for LIL results of dependent randomvariables; see [19,20] and the references therein for more details. However, it seems that the LIL in noncommutative(= quantum) probability theory has only been proved recently by Konwerska [10,11] for Hartman–Wintner’s version.Even the Kolmogorov’s LIL for independent sums in the noncommutative setting is not known. The goal of this paperis to prove Kolmogorov’s version of LIL for noncommutative martingales.Let us first recall Kolmogorov’s LIL. Let (Yn)n∈N be an independent sequence of square-integrable, centered,real random variables. Put Sn =∑ni=1 Yi and s2n = Var(Sn) =∑ni=1 E(Y 2i ). Here and in the following E denotes theexpectation and Var denotes the variance. For any x > 0, we define the notation L(x) = max{1, ln lnx}. In 1929,Kolmogorov proved that if s2n → ∞ and|Yn| ≤ αn sn√L(s2n)a.s. (1)for some positive sequence (αn) such that limn→∞ αn = 0, thenlim supn→∞Sn√s2nL(s2n)= √2 a.s. (2)Kolmogorov’s LIL for noncommutative martingales 1125Later on, Hartman–Wintner [4] proved that if (Xn) is an i.i.d. sequence of real, centered square-integrable randomvariables with variance Var(Xi) = σ 2, thenlim supn→∞Sn√nL(n)= √2σ a.s.de Acosta [2] simplified the proof of Hartman–Wintner. To compare the two results, if the sequence (Yn) are i.i.d. anduniformly bounded, then the two results coincide. Apparently, Hartman–Wintner’s LIL does not contain Kolmogorov’sversion as a special case. However, Kolmogorov’s LIL can be used in a truncation procedure to prove other LIL results;see, e.g., [16].Kolmogorov’s LIL was generalized to martingales by Stout [15]. Let (Xn,Fn)n≥1 be a martingale with E(Xn) = 0.Let Yn = Xn − Xn−1 for n ≥ 1,X0 = 0 be the associated martingale differences. Put s2n =∑ni=1 E[Y 2i |Fi−1]. ThenStout proved that if s2n → ∞ and (1) holds, then lim supn→∞ Xn/√s2nL(s2n) =√2 a.s.To state our main results, let us set up the noncommutative framework. Throughout this paper, we consider anoncommutative probability space (N , τ ). Here N is a finite von Neumann algebra and τ a normal faithful tracialstate, i.e., τ(xy) = τ(yx) for x, y ∈ N . For 1 ≤ p < ∞, define ‖x‖p = [τ(|x|p)]1/p and ‖x‖∞ = ‖x‖ for x ∈ N .In this paper ‖ · ‖ will always denote the operator norm. The noncommutative Lp space Lp(N , τ ) (or Lp(N ) forshort) is the completion of N with respect to ‖ · ‖p . τ -measurable operators affiliated to (N , τ ) are also callednoncommutative random variables; see [3,17] for more details on the measurability and noncommutative Lp spaces.Let (Nk)k=1,2,... ⊂ N be a filtration of von Neumann subalgebras with conditional expectation Ek :N → Nk . ThenEk(1) = 1 and Ek(axb) = aEk(x)b for a, b ∈ Nk and x ∈ N . It is well known that Ek extends to contractions onLp(N , τ ) for p ≥ 1; see [7].Following [10], a sequence (xn) of τ -measurable operators is said to be almost uniformly bounded by a constantK ≥ 0, denoted by lim supn→∞ xn ≤a.u.K , if for any ε > 0 and any δ > 0, there exists a projection e with τ(1 − e) < εsuch thatlim supn→∞‖xne‖ ≤ K + δ; (3)and (xn) is said to be bilaterally almost uniformly bounded by a constant K ≥ 0, denoted by lim supn→∞ xn ≤b.a.u.K ,if (3) is replaced bylim supn→∞‖exne‖ ≤ K + δ.Clearly, lim supn→∞ xn ≤a.u.K implies lim supn→∞ xn ≤b.a.u.K .For a τ -measurable operator x and t > 0, the generalized singular numbers [3] are defined byμt(x) = inf{s > 0: τ(1(s,∞)(|x|))≤ t}.In this paper, we use 1A(a) to denote the spectral projection of an operator a on the Borel set A. According to [10],a sequence of operators (xi) is said to be uniformly bounded in distribution by an operator y if there exists K > 0such that supi μt (xi) ≤ Kμt/K(y) for all t > 0. Let (xn) be a sequence of mean zero self-adjoint independent randomvariables. Konwerska [11] proved that if (xn) is uniformly bounded in distribution by a random variable y such thatτ(|y|2) = σ 2 < ∞, thenlim supn→∞1√nL(n)n∑i=1xi ≤b.a.u.Cσ.Note that if the sequence (xn) is i.i.d., which is the case in the original version of Hartman–Wintner’s LIL, then (xn)is uniformly bounded in distribution by x1. Essentially, the condition of uniform boundedness in distribution requiresthe sequence to be almost identically distributed.Our main result is an extension of Stout’s result to the noncommutative setting. Let (xn)n≥0 be a noncom-mutative self-adjoint martingale with x0 = 0 and di = xi − xi−1 the associated martingale differences. Defines2n = ‖∑ni=1 Ei−1(d2i )‖∞ and un = [L(s2n)]1/2.1126 Q. ZengTheorem 1. Let 0 = x0, x1, x2, . . . be a self-adjoint martingale in (N , τ ). Suppose s2n → ∞ and ‖dn‖∞ ≤ αnsn/unfor some sequence (αi) of positive numbers such that αn → 0 as n → ∞. Thenlim supn→∞xnsnun≤a.u.2.So far as we know, this is the first result on the LIL for noncommutative martingales. A natural question is to askfor the lower bound of LIL. As observed in [10], however, one can only expect an upper bound for LIL in the generalnoncommutative setting. Indeed, consider a free sequence of semicircular random variables (xn) (the so-called freeGaussian random variables [18]) such that the law of xn is γ0,2 (in notation, xn ∼ γ0,2) for all n. Here γ0,2 has densityfunction p(x) = 12π√4 − x2 for −2 ≤ x ≤ 2. Then it is well known in free probability theory that1√nn∑i=1xi ∼ γ0,2.It follows that limn→∞∑ni=1 xi/√nL(n) = 0 in the norm topology since a random variable with law γ0,2 is bounded.Therefore there is no reasonable notion of the positive LIL lower bound for the free semicircular sequence. Com-paring our LIL results with classical ones, we lose a constant of√2. However, since there is no hope to obtain anLIL lower bound in the general noncommutative theory, we are more interested in the order of the fluctuation forgeneral noncommutative martingales. It is also commonly acknowledged that going from the commutative theory tothe noncommutative setting usually requires considerably more technologies [14]. Due to these reasons, it seems fairto have the constant 2 in the noncommutative martingale setting.We will recall some preliminary facts in Section 2. The main result will be proved in Section 3. We will discusssome further questions in Section 4.2. PreliminariesIn this section, we give some basic definitions and collect some preliminary facts. Let us recall the vector valuednoncommutative Lp spaces for 1 ≤ p ≤ ∞ introduced by Pisier [13] and Junge [6]. Let (xn) be a sequence in Lp(N )and define∥∥(xn)∥∥Lp(∞) = inf{‖a‖2p‖b‖2p: xn = aynb,‖yn‖∞ ≤ 1}.Then Lp(∞) is defined to be the closure of all sequences with ‖(xn)‖Lp(∞) < ∞. It was shown in [8] that if everyxn is self-adjoint, then∥∥(xn)∥∥Lp(∞) = inf{‖a‖p: a ∈ Lp(N ), a ≥ 0,−a ≤ xn ≤ a for all n ∈N}.Similarly, Junge and Xu introduced in [8] the space Lp(c∞) with norm∥∥(xi)i∈I∥∥Lp(c∞)= inf{‖a‖p: a ∈ Lp(N ), a ≥ 0,−a ≤ x∗i xi ≤ a for all i ∈ I}= inf{‖b‖p: xi = yib,‖yi‖∞ ≤ 1 for all i ∈ I}.The following result is the noncommutative asymmetric version of Doob’s maximal inequality proved by Junge[6]. We add a short proof to elaborate on the constant which is implicit in the original paper.Theorem 2. Let 4 ≤ p ≤ ∞. Then, for any x ∈ Lp(N ), there exists b ∈ Lp(N ) and a sequence of contractions(yn) ⊂N such that‖b‖p ≤ 22/p‖x‖p and Enx = ynb, for all n ≥ 0.Kolmogorov’s LIL for noncommutative martingales 1127Proof. This follows from [6], Corollary 4.6. Indeed, setting r = p ≥ 4 and q = ∞, we find Enx = aznb for a, zn ∈Nand b ∈ Lp(N ). Let yn = azn/‖azn‖ ∈ N and b′ = ‖azn‖b ∈ Lp(N ). Then (yn) is a sequence of contractions,Enx = ynb′, and∥∥b′∥∥p≤ ‖a‖∞‖b‖p supn‖zn‖∞ ≤ c(p, q, r)‖x‖p,where c(p, q, r) ≤ c1/2q/(q−2)c1/2r/(r−2) = c1/21 c1/2p/(p−2) and cp is the constant in the dual Doob’s inequality. Note that1 ≤ p/(p − 2) ≤ 2. By Lemma 3.1 and Lemma 3.2 of [6], we find that cp ≤ 22(p−1)/p for 1 ≤ p ≤ 2. It follows thatc(p, q, r) ≤ 22/p . Suppose (xi)m≤i≤n is a martingale in Lp(N ). According to Theorem 2, there exist b ∈ Lp(N ) and contractions(yi)m≤i≤n ⊂N such that xi = yib for m ≤ i ≤ n and ‖b‖p ≤ 22/p‖xn‖p for p ≥ 4. It follows that∥∥(xi)m≤i≤n∥∥Lp(c∞) ≤ 22/p‖xn‖p.Doob’s inequality will be used in this form in the proof of our main result.Our proof of LIL for martingales relies on the following exponential inequality proved in [9]. Its proof was basedon Oliveira’s approach to the matrix martingales [5].Lemma 3. Let (xk) be a self-adjoint martingale with respect to the filtration (Nk,Ek) and dk = xk − xk−1 be theassociated martingale differences such that(i) τ(xk) = x0 = 0; (ii) ‖dk‖ ≤ M ; (iii) ∑nk=1 Ek−1(d2k ) ≤ D21.Thenτ(eλxn)≤ exp[(1 + ε)λ2D2]for all ε ∈ (0,1] and all λ ∈ [0,√ε/(M + Mε)].Another important tool in our proof is a noncommutative version of Borel–Cantelli lemma. To state this result, werecall from [10] that for a self-adjoint sequence (xi)i∈I of random variables, the column version of tail probability isby definitionProbc(supi∈I‖xi‖ > t)= inf{s > 0: ∃ a projection e with τ(1 − e) < s and ‖xie‖∞ ≤ t for all i ∈ I}for t > 0. It is immediate thatProbc(supi∈I‖xi‖ > t)≤ Probc(supi∈I‖xi‖ > r)(4)for t ≥ r and that if ai ≥ 1 for i ∈ I , thenProbc(supi∈I‖xi‖ > t)≤ Probc(supi∈I‖aixi‖ > t). (5)Using the notation Probc , we state two lemmas which are taken from [10].Lemma 4 (Noncommutative Borel–Cantelli lemma). Let ⋃n In = {n ∈ N: n ≥ n0} for some n0 ∈ N and (zn) be asequence of self-adjoint random variables. If for any δ > 0,∑n≥n0Probc(supm∈In‖zm‖ > γ + δ)< ∞,thenlim supn→∞zn ≤a.u.γ .1128 Q. ZengLemma 5 (Noncommutative Chebyshev inequality). Let (xi)i∈I be a self-adjoint sequence of random variables.For t > 0 and 1 ≤ p < ∞,Probc(supn‖xn‖ > t)≤ t−p‖x‖pLp(c∞).3. Law of the iterated logarithmAccording to [1], the original proof of Kolmogorov’s LIL is comparably expensive as that of Hartman–Wintner.However, our proof of Kolmogorov’s LIL here seems to be relatively easier than (the upper bound of) Hartman–Wintner’s version for the commutative case due to the exponential inequality (Lemma 3).Proof of Theorem 1. Let η ∈ (1,2) be a constant which we will determine later. To avoid annoying subscripts, wewrite s(ki) = ski in the following. Using the stopping rule in [15], we define k0 = 0 and for n ≥ 1,kn = inf{j ∈N: s2j+1 ≥ η2n}.Then s2kn+1 ≥ η2n and s2kn < η2n. Note that given ε′ > 0 there exists N1(ε′) > 0 such that for n > N1(ε′),s2kn+1u2kn+1/(s(kn+1)2u(kn+1)2)≥ η−2 ln lnη2n/ ln lnη2(n+1) ≥ (1 − ε′)2η−2.Then smum ≥ (1 − ε′)η−1s(kn+1)u(kn+1) for kn < m ≤ kn+1. For any δ′ > 0, we can find δ, ε′ > 0 and η ∈ (1,2)such that 1 + δ′ > η(1 + δ)(1 − ε′)−1. Fix β > 0 which will be determined later. Using the notation Probc with orderrelations (4) and (5), we have for n > N1(ε′)Probc(supkn<m≤kn+1∥∥∥∥ xmsmum∥∥∥∥> β(1 + δ′))≤ Probc(supkn<m≤kn+1∥∥∥∥ λxms(kn+1)u(kn+1)∥∥∥∥> λβ(1 + δ)). (6)By Lemma 5 and Theorem 2, we have for p ≥ 4,Probc(supkn<m≤kn+1∥∥∥∥ λxms(kn+1)u(kn+1)∥∥∥∥> λβ(1 + δ))≤ (λβ(1 + δ))−p∥∥∥∥(λxms(kn+1)u(kn+1))kn<m≤kn+1∥∥∥∥pLp(c∞)≤ (λβ(1 + δ))−p(22/p)p∥∥∥∥ λx(kn+1)s(kn+1)u(kn+1)∥∥∥∥pp.Using the elementary inequality |u|p ≤ ppe−p(eu + e−u), functional calculus and Lemma 3 with M = α(kn+1)×s(kn+1)/u(kn+1), D2 = s(kn+1)2, we find∥∥∥∥ λx(kn+1)s(kn+1)u(kn+1)∥∥∥∥pp≤ ppe−pτ(exp(λx(kn+1)s(kn+1)u(kn+1))+ exp(− λx(kn+1)s(kn+1)u(kn+1)))≤ 2(pe)pexp((1 + ε)λ2u(kn+1)2)Kolmogorov’s LIL for noncommutative martingales 1129provided 0 ≤ λ ≤√εu(kn+1)2(1+ε)α(kn+1) and 0 < ε ≤ 1. Hence we obtainProbc(supkn<m≤kn+1∥∥∥∥ λxms(kn+1)u(kn+1)∥∥∥∥> λβ(1 + δ))≤ 8(pλβ(1 + δ)e)pexp((1 + ε)λ2u(kn+1)2).Now optimizing in p gives p = λβ(1 + δ) and thus,Probc(supkn<m≤kn+1∥∥∥∥ λxms(kn+1)u(kn+1)∥∥∥∥> λβ(1 + δ))≤ 8 exp((1 + ε)λ2u(kn+1)2− β(1 + δ)λ).Put λ = β(1 + δ)u(kn+1)2/(2(1 + ε)). Since αn → 0, for any ε > 0, there exists N2 > 0 such that for n > N2,0 < α(kn+1) ≤ 2√εβ(1+δ) , which ensures that we can apply Lemma 3. This also implies p ≥ 4 for large n. It follows thatProbc(supkn<m≤kn+1∥∥∥∥ λxms(kn+1)u(kn+1)∥∥∥∥> λβ(1 + δ))≤ (ln s(kn+1)2)−β2(1+δ)2/(4(1+ε)).Notice that s(kn+1)2 ≥ s(kn + 1)2 ≥ η2n. Setting β = 2 in the beginning of the proof, we haveProbc(supkn<m≤kn+1∥∥∥∥ λxms(kn+1)u(kn+1)∥∥∥∥> λβ(1 + δ))≤ [(2 lnη)n]−(1+δ)2/(1+ε).By choosing ε small enough so that (1 + δ)2/(1 + ε) > 1, we find that for n0 = max{N1,N2},∑n≥n0Probc(supkn<m≤kn+1∥∥∥∥ λxms(kn+1)u(kn+1)∥∥∥∥> λβ(1 + δ))< ∞.Then (6) and Lemma 4 give the desired result. 4. Further questionsWithout the growing condition on martingale differences dn, Stout proved Hartman–Wintner’s LIL in [16] under theadditional assumption that the martingale differences are stationary ergodic. At the time of this writing, it is stillnot clear to us whether a “genuine” version (i.e., it does not satisfy Kolmogorov’s growing condition) of Hartman–Wintner’s LIL is possible for noncommutative martingales. It would be interesting to see such a result in the future.AcknowledgementsThe author would like to thank Marius Junge for inspiring discussions, Tianyi Zheng for the help on relevant literature,and Małgorzata Konwerska for sending him the preprint [11]. He is also grateful to the anonymous referee for carefulreading and suggestions on improving the paper.References[1] H. Bauer. Probability Theory. de Gruyter Studies in Mathematics 23. Walter de Gruyter, Berlin, 1996. Translated from the fourth (1991)German edition by Robert B. Burckel and revised by the author. MR13854601130 Q. Zeng[2] A. de Acosta. A new proof of the Hartman–Wintner law of the iterated logarithm. Ann. Probab. 11 (2) (1983) 270–276. MR0690128[3] T. Fack and H. Kosaki. 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