The convergence rate in noncommutative probability space have been established by several authors, e.g., Jajte [6], G¨otze and Tikhomirov [5], Chistyakov and G¨otze [3] and Stoica [13]. In particular, the authors in [3] gave estimates of the Lévy distance for freely independent partial sums and the author in [13] proved the Baum and Katz theorem in noncommutative Lorentz spaces. In this paper, we present some results on convergence rate for sequences of measurable operators under various conditions.
Trường Đại học Vinh Tạp chí khoa học, Tập 49 - Số 1A/2020, tr 51-59 CONVERGENCE RATE FOR SEQUENCES OF MEASURABLE OPERATORS IN NONCOMMUTATIVE PROBABILITY SPACE Do The Son (1) , Nguyen Van Quang (2) , Huynh Anh Thi (3) , Faculty of Fundamental Science, Industrial University of Ho Chi Minh City Department of Mathematics, Vinh University, Nghe An Province Department of Natural Sciences, Duy Tan University, Da Nang City Received on 17/10/2019, accepted for publication on 9/01/2020 Abstract: In this paper, we study the convergence rate for sequences of measurable operators under various conditions Keyword: Convergence rate; measurable operator; von Neumann algebra Introduction As it is well known, the law of large numbers (LLNs) is an essential theory in probability, statistics and related fields In noncommutative probability, this issue was considered by some authors Batty [2], Jaite [6], and Luczak [9] proved some weak and strong laws of large numbers for sequences of successively independent measurable operators Recently, Quang et al [11] presented some strong laws of large numbers for sequences of positive measurable operators and applications Other versions of LLNs can be found in [Quang et al [12], Choi et al [4], Klimczak [7]] and the references cited therein The convergence rate in noncommutative probability space have been established by several authors, e.g., Jajte [6], Găotze and Tikhomirov [5], Chistyakov and Găotze [3] and Stoica [13] In particular, the authors in [3] gave estimates of the Lévy distance for freely independent partial sums and the author in [13] proved the Baum and Katz theorem in noncommutative Lorentz spaces In this paper, we present some results on convergence rate for sequences of measurable operators under various conditions Preliminaries Let A be a von Neumann algebra (with unit 1) on a Hilbert space H and τ be a faithful normal tracial state on A A densely defined closed operator X in H is said to be affiliated to the von Neumann algebra A if U and the spectral projections of |X| belong to A, where X = U |X| is the polar decomposition of X and |X| = (X ∗ X)1/2 We notate A for the set of operators which affiliated to the von Neumann algebra A An element of A is called a measurable operator For notational consistency, A will be denoted by L0 (A, τ ) Then we have natural inclusions: A ≡ L∞ (A, τ ) ⊂ Lq (A, τ ) ⊂ Lp (A, τ ) ⊂ ⊂ L0 (A, τ ) = A 1) Email: dotheson.iuh@gmail.com (D T Son) 51 D T Son, N V Quang, H A Thi / Convergence rate for sequences of measurable operators for ≤ p ≤ q < ∞, where Lp (A, τ ) is a Banach space of all elements in L0 (A, τ ) satisfying ||X||p = [τ (|X|p )] p < ∞ For a set S of densely defined closed operators in H, W ∗ (S) denotes the smallest von Neumann algebra to which each element of S is affiliated For the case of S = {X} with a densely defined closed operator X, we write W ∗ (X) ≡ W ∗ (S) for simple notation W ∗ (X) is said to be the von Neumann algebra generated by X Denote eB (X) by the spectral projection of the self-adjoint operator X corresponding to a Borel subset B of the real line R For two self-adjoint elements X and Y in L0 (A, τ ), we say that X and Y are identically distributed if τ (eB (X)) = τ (eB (Y )) for any Borel subset B of R Let A1 and A2 be subalgebras of A Then we say that A1 and A2 are independent if τ (XY ) = τ (X)τ (Y ), ∀X ∈ A1 , ∀Y ∈ A2 Two elements X, Y ∈ L0 (A, τ ) are said to be independent if the von Neumann algebras W ∗ (X) and W ∗ (Y ) generated by X and Y, respectively, are independent A sequence {Xn , n ≥ 1} ⊂ L0 (A, τ ) is said to be pairwise independent if, for all m, n ∈ N and m = n, the algebras W ∗ (Xm ) and W ∗ (Xn ) are independent A sequence {Xn , n ≥ 1} ⊂ L0 (A, τ ) is said to be successively independent if, for every n, the algebras W ∗ (Xn ) and W ∗ (X1 , X2 , , Xn−1 ) are independent It is easily that the successively independence implies the pairwise independence Let {Xn , n ≥ 1} be a sequence in L0 (A, τ ) and X ∈ L0 (A, τ ) We say that the sequence τ {Xn , n ≥ 1} converges in measure to X, denoted by Xn − → X as n → ∞ if, for any > 0, τ e( ,∞) (|Xn − X|) → as n → ∞ For further information about the theory of noncommutative probability we refer to (Jajte [6], Nelson [10], Yeadon [15]) For convenience, from now until the end of the paper, the symbol C will denote a generic constant (0 < C < ∞) which is not necessarily the same one in each appearance Main results In this section we establish some results on convergence rate for sequence of measurable operators The following theorem is a noncommutative version of Proposition 2.4 in Li and Hu [8] Theorem 3.1 Let s > and let {Xn , n ≥ 1} be a sequence of pairwise independent measurable operators satisfying ∞ n=1 52 τ |Xn − τ (Xn )|2 ns < ∞ (3.1) Trường Đại học Vinh Tạp chí khoa học, Tập 49 - Số 1A/2020, tr 51-59 n Put Sn = Xk , then for any ε > 0, k=1 ∞ n1−s τ e[ε;∞) n=1 Sn − τ (Sn ) n < ∞ (3.2) If {Xn , n ≥ 1} is a sequence of successively independent measurable operators satisfying (3.1), then for any ε > 0, there exists a sequence of projections qn in A such that ∞ n1−s τ (qn ) < ∞, and Sn − τ (Sn ) (1 − qn ) n=1 ∞ ≤ nε (3.3) Proof For any ε > 0, by Chebyshev’s inequality and (3.1), we get ∞ n1−s τ e[ε;∞) n=1 Sn − τ (Sn ) n ∞ n1−s τ e[nε,∞) (|Sn − τ (Sn )|) = n=1 ∞ n1−s ≤ n=1 = = = ε2 ε2 ε2 ∞ n=1 n ∞ τ (|Sn − τ (Sn )|) (nε)2 n τ (Xk − τ (Xk )) 1+s k=1 n n=1 ∞ n1+s τ |Xk − τ (Xk )|2 k=1 ∞ τ |Xk − τ (Xk )|2 k=1 n=k τ |Xk − τ (Xk )|2 ∞ ≤C ks k=1 n1+s < ∞ Hence (3.1) holds Since {Xn , n ≥ 1} is a sequence of successively independent measurable operators, by Kolmogorov’s inequality, we have, for any ε > 0, there exists a sequence of projections qn in A such that τ (qn ) ≤ (nε)2 n τ |Xk − τ (Xk )|2 , k=1 and Sn − τ (Sn ) (1 − qn ) Thus, ∞ n n=1 1−s τ (qn ) < ε ∞ n=1 ns+1 ∞ ≤ nε n τ |Xk − τ (Xk )|2 < ∞ k=1 53 D T Son, N V Quang, H A Thi / Convergence rate for sequences of measurable operators The following theorem is an extension of Lemma 2.1 from Bai, Chen and Sung [1] to noncommutative probability Theorem 3.2 Let ≤ p ≤ 2, α ≤ and let {Xn , n ≥ 1} be a sequence of pairwise ∞ nα−p+1 τ (|Xn |p ) < ∞ Then for all independent measurable operators with >0 n=1 ∞ α n τ e[ n ,∞) n=1 n Xk − τ (Xk ) < ∞ k=1 Proof For each n ≥ 1, put (n) Xk = Xk e[0,n) (|Xk |) , Sn = n n Sn = n Xk , k=1 n (n) Xk , Mn = τ (Sn ), Mn = τ (Sn ) k=1 Then, for any γ > 0, we have n p ≡ e[2γ,∞ (|Sn − Mn |) ∧ e[0,γ) Sn − Mn e[0,n) (|Xk |) = ∧ k=1 Indeed, if there exists h of norm one, h ∈ p(H), then h ∈ e[0,n) (|Xk |) (H) and, con(n) sequently, Xk (h) = Xk e[0,n) (|Xk |) (h) = Xk (h), for all k = 1, 2, , n, which yields Sn (h) = Sn (h), and Mn (h) = Mn (h) Thus, from the elementary properties of the spectral decomposition, we obtain 2γ = 2γ||h||∞ ≤ |Sn − Mn | e[2γ,∞) (|Sn − Mn |) (h) ≤ (Sn − Sn )(h) = (Sn − Mn )(h) ∞ + (Sn − Mn )(h) ∞ ∞ = (Sn − Mn ) (h) ∞ + (Mn − Mn )(h) ∞ ∞ = |Sn − Mn |e[2γ,∞) |Sn − Mn | (h) ∞ ≤ γ||h||∞ = γ, which is impossible, so p = and this implies n e[2γ,∞) (|Sn − Mn |) ≺ e[γ,∞) Sn − M n ∨ e[n,∞) (|Xk |) k=1 Using the pairwise independence of the sequence {Xn , n ≥ 1} and Chebyshev’s inequal54 Trường Đại học Vinh Tạp chí khoa học, Tập 49 - Số 1A/2020, tr 51-59 ity, we obtain that n τ e[2γ,∞) (|Sn − Mn |) ≤ τ e[γ,∞) (|Sn − Mn |) + τ e[n,∞) (|Xk |) k=1 ≤ τ |Sn − Mn |2 + γ2 ≤ γ ≤ Now, take any γ2 n n τ e[n,∞) (|Xk |) k=1 n (n) τ (|Xk |2 ) + τ e[n,∞) (|Xk |) k=1 n k=1 n (n) n−p τ |Xk |p τ (|Xk |2 ) + k=1 k=1 n , we get > 0, with γ = τ e[ ,∞) (|Sn − Mn |) ≤ 2 n n n (n) τ (|Xk |2 ) k=1 n−p τ |Xk |p + k=1 Since n (n) τ |Xk |2 = λ2 τ edλ (|Xk |) n λp λ2−p τ edλ (|Xk |) = ∞ ≤ n2−p λp τ edλ (|Xk |) = n2−p τ (|Xk |p ) We have τ e[ n2 ,∞) (|Sn − Mn |) ≤ n n n n−p τ |Xk |p ≤ C n2−p τ (|Xk |p ) + k=1 k=1 n−p τ |Xk |p , k=1 which implies that ∞ ∞ α n τ e[ ,∞) (|Sn − Mn |) ≤ C n=1 n n α−p n=1 ∞ n τ |Xk |p k=1 k α−p+1 τ |Xk |p ≤C n=1 k=1 (Because n α−p ≤k α−p+1 , for all ≤ k ≤ n) ∞ k α−p+1 τ |Xk |p < ∞ ≤C k=1 55 D T Son, N V Quang, H A Thi / Convergence rate for sequences of measurable operators In Theorem 3.2, if we put α = −1, then we have the following corollary which is a noncommutative version of Lemma 2.1 in Bai, Chen and Sung [1] Corollary 3.3 Let ≤ p ≤ and let {Xn , n ≥ 1} be a sequence of pairwise independent ∞ n−p τ (|Xn |p ) < ∞ Then for all measurable operators with >0 n=1 ∞ n −1 τ e[ n ,∞) n=1 n Xk − τ (Xk ) < ∞ k=1 Taking α = in Theorem 3.2, we have the following corollary which is connected with the study of weak law of large numbers (see Corollary 3.5) Corollary 3.4 Let ≤ p ≤ and let {Xn , n ≥ 1} be a sequence of pairwise independent ∞ n−p+1 τ (|Xn |p ) < ∞ Then for all measurable operators with >0 n=1 ∞ τ e[ ,∞) n=1 n n Xk − τ (Xk ) < ∞ (3.4) k=1 Corollary 3.5 Let ≤ p ≤ and let {Xn , n ≥ 1} be a sequence of pairwise independent ∞ n−p+1 τ (|Xn |p ) < ∞ Then for all measurable operators with >0 n=1 n n τ Xk − τ (Xk ) − → as n → ∞ k=1 Proof By (3.4), we have for any ε > 0, τ e[ n ,∞) n Xk − τ (Xk ) → as n → ∞ k=1 The following theorem is a noncommutative version of Theorem 2.1 in [14] Theorem 3.6 Let {X, Xn , n ≥ 1} be a pairwise independent sequence of identically distributed measurable operators and let {an , n ≥ 1} be a sequence of positive constants with an a0 = 0, ↑ n ∞ If τ e(an ,∞) (|X|) < ∞, then for all > 0, we have n=1 ∞ n−1 τ e(an ,∞) n=1 where Xi , Sn = i=1 56 n n Sn = i=1 Xi e[0,an ] (|Xi |) Sn − τ (Sn ) < ∞, Trường Đại học Vinh Tạp chí khoa học, Tập 49 - Số 1A/2020, tr 51-59 Proof Put Yi = Xi e[0,an ] (|Xi |), give γ > 0, we have n p ≡ e(γ,∞) |Sn − τ (Sn )| ∧ e[0, γ ] |Sn − τ (Sn )| ∧ e[0,an ] (|Xi |) =0 i=1 (the proof is the same as that of Theorem 3.2 and is omitted) This yields n e(γ,∞) |Sn − τ (Sn )| ≺ e( γ ,∞) |Sn − τ (Sn )| ∨ e(an ,∞) (|Xi |) i=1 It follows that n ≤τ e τ e(γ,∞) |Sn − τ (Sn )| ( γ2 ,∞) |Sn − τ (Sn )| τ e(an ,∞) (|Xi |) + i=1 For ε > 0, by taking γ = an ε and using Chebyshev’s inequality, we obtain that n τ e(an ε,∞) |Sn − τ (Sn )| ≤ τ e( an ε ,∞) |Sn − τ (Sn )| τ e(an ,∞) (|Xi |) + i=1 ≤ τ ε a2n Sn − τ (Sn ) n τ e(an ,∞) (|Xi |) + i=1 Hence ∞ n−1 τ e(an ,∞) Sn − τ (Sn ) ≤ n=1 ≤ := ε a2n ∞ n−1 τ Sn − τ (Sn ) n=1 ∞ n−1 a−2 n τ ε2 n=1 ∞ Sn − τ (Sn ) n n−1 + n=1 ∞ + τ e(an ,∞) (|Xi |) i=1 τ e(an ,∞) (|X|) n=1 I1 + I2 ε2 Since I2 < ∞ by the assumption, it remains to show that I1 < ∞ Using the pairwise independence of the sequence {X, Xn , n ≥ 1} , we get ∞ n ∗ ∗ τ Y − τ (Y ) + [τ (Y Y ) − τ (Y )τ (Y )] I1 = n−1 a−2 i i j n i j i n=1 i=1 ∞ n−1 a−2 n = n=1 ∞ τ i=1 n n−1 a−2 n = n=1 i=j ∞ n Yi − τ (Yi ) n n−1 a−2 n ≤ n=1 i=1 ∞ a−2 n τ |X| e[0,an ] (|X|) τ |Xi |2 e[0,an ] (|Xi |) = i=1 τ |Yi |2 n=1 57 D T Son, N V Quang, H A Thi / Convergence rate for sequences of measurable operators Noting that the condition an ↑ implies n ∞ n=i i2 ≤ a2n a2i ∞ 2i ≤ 2 n n=i Therefore, we have ∞ ∞ n a−2 n I1 ≤ n=1 ∞ τ |X|2 e[ai−1 ,ai ] (|X|) = i=1 ∞ τ |X|2 e[ai−1 ,ai ] (|X|) i=1 ∞ n=i a2n i ≤2 τ |X|2 e[ai−1 ,ai ] (|X|) ≤ iτ e[ai−1 ,ai ] (|X|) a i i=1 i=1 ∞ ≤2 τ e(ai ,∞) (|X|) < ∞ i=0 REFERENCES [1] P Bai, P Y Chen, S H Sung, “On complete convergence and the strong law of large numbers for pairwise independent random variables”, Acta Math Hungar., Vol 142, No 2, pp 502-518, 2014 [2] C J K Batty, The strong law of large numbers for states and traces of a W ∗ algebra, Z Wahrsch Verw Gebiete., Vol 48, pp 177-191, 1979 [3] G P Chistyakov, F Gă otze, “Limit theorems in free probability theory I”, Ann Probab., Vol 36, pp 54-90, 2008 [4] B J Choi, U C Ji, “Convergence rates for weighted sums in noncommutative probability space”, J Math Anal Appl., Vol 409, pp 963-972, 2014 [5] F Gotze, A Tikhomirov, “Rate of convergence to the semi-circular law”, Probab Theory Related Fields, Vol 127, pp 228-276, 2003 [6] R Jajte, Strong limit theorems in non-commutative probability, Lect Notes in Maths 1110, Springer-Verlag (Berlin-Heidelberg-NewYork), 1985 [7] K Klimczak, “Strong laws of large numbers in von Neumann algebras”, Acta Math Hungar., Vol 135, No 3, pp 236-247, 2012 58 Trường Đại học Vinh Tạp chí khoa học, Tập 49 - Số 1A/2020, tr 51-59 [8] J Li, Z -C Hu, “Toeplitz lemma, complete convergence, and complete moment convergence”, Communication in statistics - Theory and methods, Vol 46, No 4, pp 1731-1743, 2017 [9] A Luczak, “Laws of large numbers in von Neumann algebras and related results”, Studia Math., Vol 81, pp 231-143, 1985 [10] E Nelson, “Notes on non-commutative integration”, J Funct Anal., Vol 15, pp 103116 1974 [11] N V Quang, D T Son, L H Son, “The strong laws of large numbers for positive measurable operators and applications”, Statist Probab Lett., Vol 124, pp 110-120, 2017 [12] N V Quang, D T Son, L H Son, “Some Kinds of Uniform Integrability and Laws of Large Numbers in Noncommutative Probability”, J Theor Probab., Vol 31, No 2, pp 1212-1234, 2018 [13] G Stoica, “Noncommutative Baum-Katz theorems”, Statist Probab Lett., Vol 79, pp 320-323, 2018 [14] S H Sung, On the strong law of large numbers for pairwise i.i.d random variables with general moment conditions, Statist Probab Lett., Vol 83, No 9, pp 1963-1968, 2013 [15] F J Yeadon, Non-commutative Lp -spaces, Math Proc Camb Phil., Vol 77, pp 91102, 1975 TÓM TẮT TỐC ĐỘ HỘI TỤ ĐỐI VỚI DÃY CÁC TỐN TỬ ĐO ĐƯỢC TRONG KHƠNG GIAN XÁC SUẤT KHƠNG GIAO HỐN Trong báo này, chúng tơi nghiên cứu tốc độ hội tụ dãy toán tử đo với điều kiện khác Từ khóa: Tốc độ hội tụ; tốn tử đo được; đại số von Neumann 59 ... / Convergence rate for sequences of measurable operators for ≤ p ≤ q < ∞, where Lp (A, τ ) is a Banach space of all elements in L0 (A, τ ) satisfying ||X||p = [τ (|X|p )] p < ∞ For a set S of. .. V Quang, H A Thi / Convergence rate for sequences of measurable operators The following theorem is an extension of Lemma 2.1 from Bai, Chen and Sung [1] to noncommutative probability Theorem... convergence rate for sequence of measurable operators The following theorem is a noncommutative version of Proposition 2.4 in Li and Hu [8] Theorem 3.1 Let s > and let {Xn , n ≥ 1} be a sequence of pairwise