A Note on the Weak Law of Large Numbers for Free Random Variables

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1 A Note o the Weak Law of Large Numbers for Free Raom Variables Raluca Bala Uiversity of Ottawa George Stoica Uiversity of New Bruswick July 28, 2006 Abstract I this article we prove that if {X k } k are free ietically istribute raom variables with commo istributio µ, a h(), are positive costats such that the fuctio f() = h() ca be extee to (0, ) a satisfies lim f (t) µ({x; x > t}) = 0 t the k= X k/h(k) M δ0 for some costats M. This is obtaie uer certai regularity coitios impose o g, h. Keywors: weak law of large umbers, ocommutative probability theory; regularly varyig fuctios. Itrouctio A classical result i probability theory is Kolmogorov s weak law of large umbers (WLLN), which states that if S = k= X k is the partial sum of a sequece of iepeet ietically istribute raom variables with commo p istributio µ (whose first orer momet may ot exist), the S / M 0 for suitable costats M, if a oly if lim t µ({x; x > t}) = 0. t Correspoig Author. Postal aress: Uiversity of Ottawa, Departmet of Mathematics a Statistics, 585 Kig Ewar Aveue, Ottawa ON, KN 6N5, Caaa. aress: rbala@uottawa.ca Postal aress: Uiversity of New Bruswick, Departmet of Mathematical Scieces, PO Box 5050, Sait Joh NB, E2L 4L5, Caaa. aress: stoica@ubsj.ca Both authors were supporte by grats from NSERC of Caaa.

2 This fuametal limit theorem was extee by Bercovici a Pata (996) a Pata (996a) to the cotext of ocommutative probability theory, i which the classical probability space (Ω, F, P ) is replace by a ocommutative probability space (A, ϕ), a the raom variables {X k } k are self-ajoit operators affiliate with A. More precisely, Pata s (996a) WLLN states that if S = k= X k is the partial sum of a sequece of free ietically istribute raom variables {X k } k with commo istributio µ a β > /2, the S / β M δ0 for suitable costats M, if a oly if lim t t/β µ({x; x > t}) = 0. The purpose of this paper is to prove that a similar WLLN cotiues to hol for a sequece of free ietically istribute raom variables X k which are weighte by costats h(k), a whose partial sum S = k= X k/h(k) is ormalize by aother costat. This is obtaie uer the coitio (C) lim f (t) µ({x; x > t}) = 0, t where f() = h() a we assume that the fuctio f ca be extee to (0, ) (a satisfies some aitioal coitios). This lie of research was ispire by the result of Jaite (2003) i the cotext of classical probability a.s. theory, who prove that S / M 0 for suitable costats M, if a oly if (C ) f (t)µ(t) <. We begi by itroucig the termiology a otatio specific to ocommutative probability theory. A W -probability space is a pair (A, ϕ) be a probability space, where A is a complex uital vo Neuma subalgebra of some L(H) (the space of boue liear operators o a Hilbert space H) a ϕ is a ormal faithful trace. A raom variable X is a self-ajoit operator affiliate with A, i.e. u(x) A for ay boue Borel fuctio u o R. The istributio of a raom variable X is a probability measure o R give by µ X = ϕ E X, where E X is the spectral measure of X. We have ϕ(u(x)) = u(t)µ X (t) for every boue Borel fuctio u o R. A sequece of raom variables {X } coverges i istributio to a probability measure ν (a we write X ν) if µx coverges to ν weakly. Fially, the raom variables {X } are iepeet if ϕ(a j A k ) = ϕ(a j )ϕ(a k ) for every A j W (X j ), A k W (X k ), with j k, where W (X i ) eotes the smallest vo Newma to which each elemet of X i is affiliate (see p. 592, Pata, 996a). 2

3 Throughout this paper, we let {} be a oecreasig sequece of positive umbers with, {h()} a sequece of positive umbers a f() = h(). We eote by C a geeric costat, which may be ifferet from lie to lie. 2 Results a Examples Our first theorem is the ocommutative aalogue of the classical egeerate covergece criterio (see e.g. Theorem of Chug, 200, or p.278, Loève, 963). Its proof uses the same trucatio proceure as i the classical case which carries over to the ocommutative framework. Theorem 2. Let {X k } k be a sequece of iepeet raom variables i a W -probability space (A, ϕ), a eote by µ k the istributio of X k. If the (C) (C2) k= µ k ({x; x > f()}) 0 k= 2 k= f() h(k) 2 t 2 µ k (t) 0 f() ( ) f() X k tµ k (t) δ 0. h(k) f() Proof: For all k, we efie the trucate raom variables Xk, = X k E Xk ([ f(), f()]) (recallig that E Xk is the spectral measure of the operator X k ), as well as the associate first orer momets m k, = f() f() tµ k(t). Let S = k= h(k) X k, S = k= h(k) X k,, M = We wat to show that S M coverges i istributio to δ 0, i.e. k= h(k) m k,. µ S M ( ε ) 0 ε > 0 () where ε = {x; x > ε}. By Propositio 3. (Pata, 996a), µ S M ( ε ) µ S S ( 0) + µ S M ( ε ). Usig repeately Propositio 3. (Pata, 996a) a coitio (C), we get µ S S ( 0) µ h(k) (X k X k, ) ( 0 ) = k= 3 µ Xk X ( 0) k, k=

4 = µ Xk E Xk ( f() )( 0 ) = k= µ k ( f() ) 0. Usig Chebyshev s iequality, the iepeece of X k s a coitio (C2), µ S M ( ε ) ε 2 ϕ((s M ) 2 ) ε 2 2 k= This coclues the proof of (). k= ε 2 2 k= f() h(k) 2 t 2 µ k (t) 0 f() h(k) 2 ϕ((x k,) 2 ) = Whe the raom variables {X k } k have a commo istributio µ, (C) becomes: (C ) lim µ({x : x > f()}) = 0 (which is almost the same as (C)). Moreover, a result o p. 92 of Aler a Rosalsky (99) says that (C ) implies (C2) (with µ k = µ for all k), if either oe of the followig sets of coitios hol: (F ) f, (F 2) f(), A() := 2 k= f(), ( ) h(k) 2 = O h() 2 k= h(k) 2 = o(), k= f(k) 2 k 2 = O(A ) (Here the symbols u or u are use to iicate that the sequece {u } is oecreasig, respectively oicreasig.) Hece, we obtai the followig: Corollary 2.2 Let {X k } k be a sequece of iepeet raom variables i a W -probability space, with commo istributio µ. If (F )-(F 2) a (C ) hol, the k= ( ) f() X k t µ(t) h(k) f() δ 0. Example 2.3 Let = a, a 0 a h() = b, b > /2. We see that (F ), respectively (F 2) hol, epeig o whether a + b < or a + b. Later i this paper, we will improve this example by requirig oly a, b 0; a + b > /2; see Example 2.0. Example 2.4 If h is oicreasig a f()/ is oecreasig, the (F 2) is satisfie. As a example we may take = ρ, ρ > a h() = / log. 4

5 I orer to elarge the class of examples, we will use a variat of Theorem 2., which ca be obtaie for istace by applyig Theorem 2. to the raom variables X k = X k /h(k) a the sequeces =, h() =. Note that i this case X k has istributio µ k efie by µ k (B) := µ k (h(k)b) for ay Borel set B R. Theorem 2.5 Let {X k } k be a sequece of iepeet raom variables i a W -probability space, a eote by µ k the istributio of X k. If the ( C) ( C2) k= µ k ({x; x > h(k)}) 0 k= 2 k= h(k) h(k) 2 t 2 µ k (t) 0 h(k) ( ) h(k) X k t µ k (t) δ 0. h(k) h(k) Suppose ow that the raom variables X k have commo istributio µ. Our goal is to show that i this case, (C) implies both ( C) a ( C2). For this we will assume that, h() satisfy (F 3) m := if (F 4) f (h(k)) C, k= for large eough a the fuctio f ca be extee to (0, ) a satisfies: (F 5) the iverse f of f exists a satisfies lim t f (t) = (F 6) f is regularly varyig at with iex ρ > /2, i.e. for every λ > 0 lim f(λx)/f(x) = λρ x Let k, = h(k) for all k a τ(t) = f (t) µ({x; x > t}) for t > 0. Propositio 2.6 Uer (F 3)-(F 5), (C) implies ( C). Proof: Let ε > 0 be arbitrary. By (C), there exists T = T ε such that τ(t) < ε for all t T. Sice a (F 3) hols, there exists N = N ε such that k, m T for all N, k. Therefore, for every N µ({x; x > k, }) = k= k= f ( k, ) τ( k,) ε 5 k= f ( k, ) Cε,

6 where we use (F 4) for the last iequality. This coclues the proof. The followig lemma geeralizes Lemma 4.(iii) (Bercovici a Pata, 996) to the case of ivertible, regularly varyig fuctios f. Lemma 2.7 Uer (F 5)-(F 6), (C) implies that f (y) lim y y k y Proof: Usig itegratio by parts, we have y t k µ(t) = y k µ({x; x > y}) + k The result will follow oce we prove that f (y) lim y y k 0 t k µ(t) = 0, k 2. 0 t k µ({x; x > t})t. t k µ(x; x > t)t = 0. (2) Note that f is regularly varyig at with iex /ρ. Usig Karamata s Represetatio Theorem (see Theorems.3.,.4., Bigham, Golie a Teugels, 987), respectively Potter s Theorem (see Theorem.5.6.(iii), Bigham, Golie a Teugels, 987), we kow that for every δ > 0, A > there exists C = C δ > 0, Y = Y δ,a > 0 such that f (y) Cy (/ρ)+δ, y Y (3) f (y) ( y ) (/ρ)+δ f (t) A, y t Y (4) t Let 0 < δ < k (/ρ) a A > be fixe. Let ε > 0 be arbitrary. By (C), there exists N = N ε > Y such that τ(t) < ε, t > N. Usig (3), f (y) y k N 0 t k µ(x; x > t)t N k f (y) y k N k Cy (/ρ)+δ k ε (5) for y large eough. Usig (4), f (y) y k N t k µ(x; x > t)t ε f (y) y k = Aε y α N t α t = Aε α N [ t k Aɛ f t (t) y k ( ) α ] N Aε y α where α := k (/ρ) δ > 0. The proof of (2) is complete by (5)- (6). Usig the previous lemma, we obtai the followig result. N ( t k y ) (/ρ)+δ t t (6) 6

7 Propositio 2.8 Uer (F 3)-(F 6), (C) implies ( C2). Proof: Let ε > 0 be arbitrary. By Lemma 2.7, there exists Y = Y ε such that v(y) := f (y) y 2 y t 2 µ(t) ε, y Y. Sice a (F 3) hols, there exists N = N ɛ such that k, > Y for all N, k. Hece, usig (F 4) we have 2 k= k, k, k, t 2 µ(t) = k= for all N, which coclues the proof. v( k, ) f ( k, ) ε f ( k, ) Cε I summary, Theorem 2.5, Propositio 2.6 a Propositio 2.8, lea us to the followig result. Corollary 2.9 Let {X k } k be a sequece of iepeet raom variables i a W -probability space, with commo istributio µ. If (F 3)-(F 6) a (C) hol, the ( ) h(k) X k t µ(t) δ 0. h(k) k= h(k) Example 2.0 Let = a a h() = b, where a, b 0, a + b > /2. I this case f() = a+b a (F 3)-(F 6) hol: (F 3), (F 5) a (F 6) ca be checke irectly, whereas for (F 4) we have: ( a k b ) /(a+b) = a/(a+b) k b/(a+b) a/(a+b) C b/(a+b) = C. k= k= Example 2. If h is oicreasig a f is oecreasig (for x large eough), the (F 4) hols: k= f (h(k)) As a example, we may take k= k= f (f()) = = a l () a h() = + k= b l 2 () =. where a, b 0, a b > /2 a l (x), l 2 (x) are slowly varyig fuctios such that lim x l (x) = lim x l 2 (x) = a l (x)/l 2 (x) is oecreasig for 7

8 x large (e.g. l (x) = log x, l 2 (x) = log log x). I this case f() = a l () + a b l ()/l 2 () a (F 3)-(F 6) hol. I particular, by takig = ρ l() a h() = where ρ > /2 a l(x) is a slowly varyig fuctio, we elarge the class of examples cosiere by Gut (2004), where ρ >. Cocluig Remarks: (a) Corollary 2.9 extes several results i ocommutative probability theory, such as the Kolmogorov WLLN (cf. Bercovici a Pata, 996) a the Marcikiewicz WLLN (cf. Pata, 996a), a i aitio gives ew such WLLN s cosierig for istace regularly varyig weights. (b) Propositios 2.6, 2.8 show that uer (F 3) (F 6), (C) is a sufficiet coitio for the WLLN with regularly varyig ormalizig costats eve i the classical sese, which seems to be a ew result. (See Theorem.3 of Gut, 2004 which treats the case h() =.) (c) I view of the the cetral limit theorem for free raom variables (cf. Pata, 996b), the iex ρ i (F 6) has to be strictly larger tha /2. () The case of logarithmic averages (i.e. = log, h() = ) is ot covere either by coitios (F ) (F 2), or by coitios (F 3) (F 6). I the classical theory, coitio (C ) (with f(t) := t log t) is kow to be ecessary a sufficiet for the strog LLN (see Jaite, 2003). However, eve i this classical settig, it is ot clear whether the WLLN for logarithmic averages hols, uer (C) aloe. (e) The ecessity of coitio (C) for the WLLN (i Corrolary 2.9) remais a ope problem. I orer to tackle this problem, oe woul have to make use of the complex fuctio φ µ (z) = Fµ (z) z which is the ocommutative aalogue of the classical log-characteristic fuctio. Here F µ (z) = /G µ (z) with G µ (z) = /(z x)µ(x) a F µ is the iverse with respect to compositio. Refereces [] Aler, A. a Rosalsky, A. (99), O the weak law of large umbers for orme weighte sums of i.i.. raom variables, Iter. J. Math. & Math. Sci. 4, [2] Bigham, N. H., Golie, C. M. a Teugels, J. L. (987), Regular Variatio (Cambrige Uiversity Press, Cambrige). [3] Bercovici, H. a Pata, V. (996), The law of large umbers for free ietically istribute raom variables, A. Probab. 24,

9 [4] Chug, K. L. (200), A Course i Probability Theory, Thir Eitio (Acaemic Press, Sa Diego). [5] Gut, A. (2004), A extesio of the Kolmogorov-Feller weak law of large umbers with a applicatio to the St. Petersburg game, J. Theoret. Probab. 7, [6] Jaite, R. (2003), O the strog law of large umbers, A. Probab. 3, [7] Loève, M. (963), Probability Theory, Thir Eitio (D. va Nostra Compay, Priceto, New Jersey). [8] Pata, V. (996a), A geeralize weak law of large umbers for ocommutig raom variables, Iiaa Uiv. Math. J. 45, [9] Pata, V. (996b), The cetral limit theorem for free aitive covolutio, J. Fuct. Aal. 40,