Noncommutative Baum–Katz theorems

Statistics and Probability Letters 79 (2009) 320–323

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Noncommutative Baum–Katz theorems George Stoica Department of Mathematical Sciences, University of New Brunswick, Saint John NB, E2L 4L5, Canada

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Article history: Received 14 February 2008 Accepted 27 August 2008 Available online 30 August 2008

We prove two Baum–Katz results on the rate of convergence in the law of large numbers for successively independent random variables in tracial noncommutative probability spaces. © 2008 Elsevier B.V. All rights reserved.

MSC: 46L53 60F25 60F10

1. Introduction and results There are two traditional ways of estimating the rate of convergence in the law of large numbers (LLN): using the Lévy distance for the distribution functions of partial sums, and Baum–Katz moment conditions on the tail of the partial sums. Both approaches have been extensively used for the classical LLN, starting with the i.i.d. case, see Loève (1977), Baum and Katz (1965), and various extensions have been obtained over the years. In noncommutative probability, several versions of independence have been introduced and the corresponding LLN are now proved; see Batty (1979), Jajte (1985, 1991), Stoica (1993), Bercovici and Pata (1996), Lindsay and Pata (1997), etc. However, the rate of convergence in those LLNs has been completely neglected; only the very recent paper by Chistyakov and Götze (2008) gives estimates of the Lévy distance for freely independent partial sums. It is the purpose of this work to fill in the gap, in part, by obtaining Baum–Katz estimates that require only a very mild form of noncommutative independence. The main problem in this approach is the lack of a Fuk–Nagaev inequality in (any) noncommutative set-up; to circumvent this drawback, we are going to adapt the doubletruncation techniques of Erdős (1949) and Katz (1963) in the noncommutative set-up. We consider a tracial W ∗ -probability space (M , τ ) whose elements X , Y , etc. are called random variables, where M is a von Neumann algebra, and τ is a faithful normal trace with τ (1) = 1. Thus τ is a linear weak∗ -continuous functional, τ (X ∗ X ) > 0 for X 6= 0 and τ (XY ) = τ (YX ) for X , Y ∈ M. We denote by L0 (M, τ ) the ∗-algebra of random variables affiliated with M , that is, all spectral projections of |X | belong to M . If X is a self-adjoint random variable, then X can be viewed as an R∞ (in general unbounded) self-adjoint operator affiliated with M , and it can be given as a spectral integral X = −∞ teX (dt ), where the projection-valued Borel measure eX takes values in M . The distribution of X is the probability measure on R given by τ (eX (A)) for all Borel subsets A ⊆ R; its inverse its given by the generalized singular number


µt (X ) = inf s > 0 : τ e|X | ((s, ∞)) ≤ t , for t > 0. The noncommutative Lorentz spaces Mp,q (M , τ ) (see Pisier and Xu (2003)) are defined for p, q ≥ 1 as the set of measurable operators X satisfying kX kp,q < ∞, where Z ∞ 1/q  q/p−1 q  t µt (X )dt if q < ∞, kX kp,q = 0 1/p  sup t µ (X ) if q = ∞. t >0

t

E-mail address: [email protected]. 0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.08.013

G. Stoica / Statistics and Probability Letters 79 (2009) 320–323

321

Two subalgebras M1 , M2 of M are said to be independent if τ (X1 X2 ) = τ (X1 )τ (X2 ) for every Xi ∈ Mi , i = 1, 2. If M1 , M2 are independent and commute, they are said to be classically independent. Two sets of random variables are said to be independent if they generate independent algebras. A sequence X1 , X2 , . . . is said to be successively independent if the sets {X1 , X2 , . . . , Xn } and {Xn+1 } are independent for all n. In particular, sequences of classically independent random variables are successively independent. A family {Mi }i∈I of subalgebras of M are said to be free (or freely independent) if, given a natural number n and elements Xj ∈ Mij , j = 1, 2, . . . , n, such that τ (Xj ) = 0 and ij 6= ij+1 for all j, we have τ (X1 X2 . . . Xn ) = 0. A family {Xi }i∈I is said to be free if the algebras generated by Xi form a free family. In particular, a free sequence of random variables is also successively independent. Note however that classical independence is incompatible with freeness: if two variables are both free and classically independent, one of them must be a constant multiple of the identity (i.e., a constant). A family {Xi }i∈I is said to be identically distributed if the distributions τ (eXi ) are identical for all i ∈ I. Theorem 1. Let us have p ≥ 1, 1 ≤ q ≤ ∞, and (Xn )n≥1 a sequence of successively independent identically distributed random variables in Mp,q (M , τ ). Then, for any 1 ≤ r < 2 such that p ≥ r, one has ∞ X

np/r −2 τ e|X1 +X2 +···+Xn −nτ (X1 )| (n1/r , ∞)





< ∞.

(1)

n =1

The interpretation of estimate (1) is that of a rate of convergence in the noncommutative versions of the LLN, where various types of convergence have been employed; see Batty (1979), Jajte (1985, 1991), Stoica (1993), Bercovici and Pata (1996), Pata (1996), and Chistyakov and Götze (2008). In particular, Theorem 1 applies to both classical and free independent sequences in a tracial W ∗ -probability space. Estimate (1) is the best one can obtain, as is proved to be optimal in the commutative i.i.d. set-up; see Baum and Katz (1965). Also, estimate (1) does not depend on q (see the above definition of the noncommutative Lorentz spaces); thus the same rate applies to random variables in the noncommutative Kunze Lp spaces, p ≥ 1, or Haagerup Lp -spaces, p > 1 (see Pisier and Xu (2003)). In the commutative case, formula (1) is valid for 0 < r < 1 as well; we were not able to prove it in the noncommutative set-up. When the random variables are no longer identically distributed, we obtain a weaker Baum–Katz result, as follows. Theorem 2. Let us have p > 1, 1 ≤ q ≤ ∞, and (Xn )n≥1 a sequence of successively independent random variables uniformly bounded in Mp,q (M , τ ). Then, for any 1 ≤ r < 2 such that p ≥ r and t < p/r − 2, one has ∞ X

nt τ e|Y1 +Y2 +···+Yn | (n1/r , ∞)





< ∞,

(2)

n =1

where Yi = Xi − τ (Xi e|Xi | ([0, n1/r ))), i = 1, 2, . . . . Theorem 2 applies to the noncommutative weighted LLN in Lindsay and Pata (1997), Pata (1996), Chistyakov and Götze (2008), and Stoica (2000, 2007). Estimate (2) is the best one can obtain in such a set-up, as the normalization np/r −2 is proved to be no longer optimal even in the commutative weighted LLN; see Lanzinger and Stadtmüller (2003). Also notice that Theorem 2 does not apply when p = 1; indeed, one may not have even a LLN for certain weights in commutative or noncommutative L1 space; see Lindsay and Pata (1997), Lanzinger and Stadtmüller (2003) and Stoica (2007). Open problem. Some of the hypotheses in Theorems 1 and 2 may improve if one follows an alternative route in proving them, such as a noncommutative Fuk–Nagaev inequality for traces or even for states. To our knowledge, such results are not yet available. 2. Proofs We denote by C > 0 a constant whose value is not essential, does not depend on n, and may vary from line to line. Proof of Theorem 1. As r ≥ 1, without loss of generality, we may assume that τ (X1 ) = 0. Choose γ > 0 such that γ > r /2, γ > r /p and 1/2 + r /(2p) < γ < 1. Following Erdős (1949), one defines

 Zk =

Xk 0

if τ e|Xk | [0, nγ /r ] otherwise.





=1

Let us have 2i ≤ n < 2i+1 . We have



τ e|X1 +X2 +···+Xn | (n1/r , ∞) ≤ Un + Vn + Wn ,

(3)

where Un = nτ e|X1 | (2(i−2)/r , ∞) Vn =

X









τ e|Xk | (nγ /r , ∞) · τ e|Xj | (nγ /r , ∞)

k6=j

Wn = τ e|Z1 +Z2 +···+Zn | (2(i−2)/r , ∞)





.

(4)

322

G. Stoica / Statistics and Probability Letters 79 (2009) 320–323

We have Un ≤ 2i+1 τ e|X1 | (2(i−2)/r , ∞) ; hence





∞ X

np/r −2 Un =

∞ X

np/r −2 · 2i+1 · τ e|X1 | (2(i−2)/r , ∞)

X





i=0 2i ≤n<2i+1

n =1

≤

∞ X

2i · 2i+1 · (2i+1 )p/r −2 · τ e|X1 | (2(i−2)/r , ∞)





i=0

= C + 23p/r −1

∞ X

2pi/r · τ e|X1 | (2i/r , ∞)





i =0

< ∞ as kX1 kp,q < ∞.

Then we have Vn ≤ n2 τ 2 e|X1 | (nγ /r , ∞) ; hence applying Chebyshev’s inequality with p ≥ 1 (see Jajte (1985)):

τ e|X1 | (nγ /r , ∞) ≤ n−pγ /r τ (|X1 |p ),

(5)

we obtain ∞ X

np/r −2 Vn ≤ C ·

n =1

∞ X

np/r −(2pγ )/r < ∞ as γ > 1/2 + r /(2p).

(6)

n=1

As 2i−2 > n/8 we have Wn ≤ τ e|Z1 +Z2 +···+Zn | ((n/8)1/r , ∞) . Then we use the following combinatorial estimate from Katz (1963, p. 315):





τ (|Z1 + Z2 + · · · + Zn |2M ([p]+1) ) ≤ C · n2M ([p]+1)γ /r −γ p/r +1 provided M > (γ p − r )/ ((2γ − r )([p] + 1)) , where [p] is the smallest integer larger than p (here we used that γ > r /2 and γ > r /p). Using again Chebyshev’s inequality, we obtain ∞ X

np/r −2 Wn ≤ C ·

n =1

∞ X

np/r −2

n =1

τ (|Z1 + Z2 + · · · + Zn |2M ([p]+1) ) n2M ([p]+1)/r

Combining Eqs. (3)–(7) we obtain (1) and the proof is finished.

< ∞.

(7)



Proof of Theorem 2. As the sequence (Xn )n≥1 is uniformly bounded in Mp,q (M , τ ), we have

µt (Xk ) ≤ C · t 1/p for k = 1, 2, . . . and t > 0.

(8)

Indeed, formula (8) is obvious if q = ∞ and, for q finite, we observe that t −1+q/p

t

Z 0

µqs (Xk )ds ≤ sup kXk kqp,q < ∞ for k = 1, 2, . . . and t > 0. k

According to the definition of µt , formula (8) is equivalent to


τ e|Xk | ((t , ∞)) ≤ C · t −p for k = 1, 2, . . . and t > 0.

(9)

Recall the following result (cf. Lindsay and Pata (1997), Proposition 2.3) on spectral projections of the random variables U1 , U2 , . . . , Un in a tracial W ∗ -probability space: we have n
X
τ e|U | ((ε, ∞)) ≤ τ e|Uk | ((εk , ∞)) , k=1

where U = U1 + U2 + · · · + Un and ε = ε1 + ε2 + · · · + εn , with ε1 , ε2 , . . . , εn ≥ 0. The quoted result gives n



X

τ e|Y1 +Y2 +···+Yn | (n1/r , ∞) ≤ τ e|T1 +T2 +···+Tn | (n1/r , ∞) + τ e|Xk | (n1/r , ∞) ,

(10)

k=1

where Tk := Xk χ[−n1/r ,n1/r ] (Xk ) − τ Xk e|Xk | [0, n1/r ) . By Chebyshev’s inequality and using the fact that the variance operator τ (U 2 ) − τ (U )2 is linear on successively independent random variables (cf. Lindsay and Pata (1997), p. 539), the right-hand side of inequality (10) is





≤

C n2/r

( ) n n h X X
2 i 
2

τ Xk χ[−n1/r ,n1/r ] (Xk ) − τ Xk χ[−n1/r ,n1/r ] (Xk ) + τ e|Xk | (n1/r , ∞) . k=1

k=1

(11)

G. Stoica / Statistics and Probability Letters 79 (2009) 320–323

323

By Fubini’s theorem we have the following version of the variance inequality:

τ (U ) − τ (U ) ≤ 2

2

Z

∞


|a|τ e|U | ((a, ∞)) da, −∞

and hence the last expression in (11) is

≤

n Z C X

n1/r

n2/r k=1 −n1/r

n X


|a|τ e|Xk | ((a, ∞)) da + τ e|Xk | (n1/r , ∞) .

(12)

k=1

Using estimate (9), both terms in the last expression in (12) are ≤ C · n1−p/r . So, for t < p/r − 2 we have ∞ X

nt τ e|Y1 +Y2 +···+Yn | (n1/r , ∞)





∞ X

nt −1+p/r < ∞,

n =1

n =1

and Eq. (2) is proved.

≤C·



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