On the Laws of Large Numbers for Double Arrays of Independent Random Elements in Banach Spaces

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On the Laws of Large Numbers for Double Arrays of Independent Random Elements in Banach Spaces

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For a double array of independent random elements {Vmn, m ≥ 1, n ≥ 1} in a real separable Banach space, conditions are provided under which the weak and strong laws of large numbers for the double sums Pm i=1 Pn j=1 Vij , m ≥ 1, n ≥ 1 are equivalent. Both the identically distributed and the nonidentically distributed cases are treated. In the main theorems, no assumptions are made concerning the geometry of the underlying Banach space. These theorems are applied to obtain Kolmogorov, BrunkChung, and MarcinkiewiczZygmund type strong laws of large numbers for double sums in Rademacher type p (1 ≤ p ≤ 2) Banach spaces.

On the Laws of Large Numbers for Double Arrays of Independent Random Elements in Banach Spaces ∗ Andrew ROSALSKY, Le Van THANH, Nguyen Thi THUY Abstract For a double array of independent random elements {Vmn , m ≥ 1, n ≥ 1} in a real separable Banach space, conditions are provided under which the weak and strong laws of large numbers for n the double sums m i=1 j=1 Vij , m ≥ 1, n ≥ are equivalent Both the identically distributed and the nonidentically distributed cases are treated In the main theorems, no assumptions are made concerning the geometry of the underlying Banach space These theorems are applied to obtain Kolmogorov, Brunk-Chung, and Marcinkiewicz-Zygmund type strong laws of large numbers for double sums in Rademacher type p (1 ≤ p ≤ 2) Banach spaces Key Words and Phrases: Real separable Banach space; Double array of independent random elements; Strong and weak laws of large numbers; Almost sure convergence; Convergence in probability; Rademacher type p Banach space 2010 Mathematics Subject Classifications: 60F05, 60F15, 60B11, 60B12 Introduction Throughout this paper, we consider a double array {Vmn , m ≥ 1, n ≥ 1} of independent random elements defined on a probability space (Ω, F, P ) and taking values in a real separable Banach space X with norm || · || We provide conditions under which the strong law of large numbers (SLLN) and m n the weak law of large numbers (WLLN) for the double sums i=1 j=1 Vij are equivalent Such n double sums differ substantially from the partial sums i=1 Vi , n ≥ of a sequence of independent random elements {Vn , n ≥ 1} because of the partial (in lieu of linear) ordering of the index set {(i, j), i ≥ 1, j ≥ 1} We treat both the independent and identically distributed (i.i.d.) and the independent but nonidentically distributed cases In the main results (Theorems 3.1 and 3.7), no assumptions are made concerning the geometry of the underlying Banach space We then apply the main results to obtain Kolmogorov, Brunk-Chung, and Marcinkiewicz-Zygmund type SLLNs for double sums in Rademacher type p (1 ≤ p ≤ 2) Banach spaces While in the current work attention is restricted to considering double sums, the results can of course be extended by the same method to multiple sums over lattice points of any dimension ∗ The research of the second author was supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM) and the Vietnam National Foundation for Sciences and Technology Development (NAFOSTED) under grant number 101.01.2012.13 The research of the third author was supported by the Vietnam National Foundation for Sciences and Technology Development (NAFOSTED) under grant number 101.03.2012.17 The reader may refer to Rosalsky and Thanh [1] for a brief discussion of a historical nature concerning double sums and on their importance in the field of statistical physics For the case of i.i.d real valued random variables, a major surrey article concerning double sums was prepared by Pyke [2] In Pyke [2], he discussed fluctuation theory, the limiting Brownian sheet, the SLLN, and various other limit theorems Currently, Professor Oleg I Klesov (National Technical University of Ukraine) is preparing a comprehensive book on multiple sums of independent random variables The plan of the paper is as follows Notation, technical definitions, and five known lemmas which are used in proving the main results are consolidated into Section In Section 3, we establish the main results after first proving three new lemmas The applications of the main results are presented in Section Section contains an example pertaining to Theorems 3.1 and 4.1 Preliminaries In this section, notation, technical definitions, and lemmas which are needed in connection with the main results will be presented For a, b ∈ R, min{a, b} and max{a, b} will be denoted, respectively, by a∧b and a∨b Throughout this paper, the symbol C will denote a generic constant (0 < C < ∞) which is not necessarily the same one in each appearance The expected value or mean of an X -valued random element V , denoted EV , is defined to be the Pettis integral provided it exists If E V < ∞, then (see, e.g., Taylor [3], p 40) V has an expected value But the expected value can exist when E V = ∞ For an example, see Taylor [3], p 41 Hoffmann-Jørgensen and Pisier [4] proved for ≤ p ≤ that a real separable Banach space is of Rademacher type p if and only if there exists a constant C such that p n n E||Vj ||p ≤C Vj E j=1 (2.1) j=1 for every finite collection {V1 , , Vn } of independent mean random elements For the double array of random elements {Vmn , m ≥ 1, n ≥ 1}, we write m n Vij , m ≥ 1, n ≥ S(m, n) = Smn = i=1 j=1 For sums of independent random elements, the first lemma provides in (2.2) and (2.3) a MarcinkiewiczZygmund type inequality and a Rosenthal type inequality, respectively Lemma 2.1 is due to de Acosta [5, Theorem 2.1] Lemma 2.1 Let {Vj , ≤ j ≤ n} be a collection of n independent random elements Then for every p ≥ 1, there is a positive constant Cp < ∞ depending only on p such that n n Vj − E E j=1 and n j=1 ≤ Cp j=1 n Vj − E E n p Vj j=1 p , for ≤ p ≤ 2, (2.2) j=1 n p Vj E Vj n ≤ Cp E Vj j=1 p/2 + E Vj p , for p > j=1 The following lemma is due to Hoffmann-Jørgensen [6]; see the proof of Theorem 3.1 of [6] (2.3) Lemma 2.2 Let {Vj , ≤ j ≤ n} be a collection of n independent symmetric random elements in a real separable Banach space Then for all t > 0, s > n n Vj > 2t + s) ≤ 4P ( P( j=1 Vj > t) + P ( max Vj > s) 1≤j≤n j=1 The next lemma is L´evy’s inequality for double arrays of independent symmetric random elements in Banach spaces It is due to Etemadi [7, Corollary 1.2] We note that Etemadi [7] established the result for d-dimensional arrays where d is arbitrary positive integer Lemma 2.3 Let {Vij , ≤ i ≤ m, ≤ j ≤ n} be a collection of mn independent symmetric random elements in a real separable Banach space Then there exist a constant C such that for all t > 0, P ( max k≤m,l≤n Skl > t) ≤ CP ( Smn > t/C) The following result is a double sum analogue of the Toeplitz lemma (see, e.g., Lo`eve [8], p 250) and is due to Stadtmă uller and Thanh [9, Lemma 2.2] Lemma 2.4 Let {amnij , ≤ i ≤ m + 1, ≤ j ≤ n + 1, m ≥ 1, n ≥ 1} be an array of positive constants such that m+1 n+1 amnij ≤ C and sup m≥1,n≥1 i=1 j=1 lim m∨n→∞ amnij = for every fixed i, j If {xmn , m ≥ 1, n ≥ 1} is a double array of constants with lim m∨n→∞ xmn = 0, then m+1 n+1 amnij xij = lim m∨n→∞ i=1 j=1 The last lemma in this section has an easy proof; see Rosalsky and Thanh [10, Lemma 2.1] Lemma 2.5 Let {Vmn , m ≥ 1, n ≥ 1} be a double array of random elements in a real separable Banach space and let p > If ∞ ∞ E Vmn p < ∞, m=1 n=1 then Vmn → almost surely (a.s.) and in Lp as m ∨ n → ∞ Finally, we note that the Borel-Cantelli lemma (both the convergence and divergence halves) carries over to an array of events {Amn , m ≥ 1, n ≥ 1} since the sets {(m, n) : m ≥ 1, n ≥ 1} and {k : k ≥ 1} are in one-to-one correspondence with each other Of course, for the divergence half, it is assumed that the array {Amn , m ≥ 1, n ≥ 1} is comprised of independent events 3 Main Results With the preliminaries accounted for, the first main result may be established Theorem 3.1 considers the independent but nonidentically distributed case while Theorem 3.7 considers the i.i.d case In these theorems, no assumptions are made concerning the geometry of the underlying Banach space In Theorem 3.1, condition (3.1) is a Kolmogorov type condition for the SLLN for double arrays whereas condition (3.4) is a Brunk-Chung type condition for the SLLN for double arrays Theorem 3.1 Let α > 0, β > and let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent random elements in a real separable Banach space (i) Assume that ∞ ∞ E Vmn p < ∞ for some ≤ p ≤ (3.1) mαp nβp m=1 n=1 Then Smn P → as m ∨ n → ∞ m α nβ (3.2) Smn → a.s as m ∨ n → ∞ mα nβ (3.3) E Vmn 2p 2αp+1−p m n2βp+1−p (3.4) if and only if (ii) Assume that ∞ ∞ m=1 n=1 < ∞ for some p > Then (3.2) and (3.3) are equivalent The proof of Theorem 3.1 has several steps so we will break it up into three lemmas Some of the lemmas may be of independent interest The first lemma ensures that in Theorem 3.1, it suffices to assume that the array {Vmn , m ≥ 1, n ≥ 1} is comprised of symmetric random elements Lemma 3.2 is a double sum analogue (with more general norming constants) of Lemma of Etemadi [11] Lemma 3.2 Let α > 0, β > and let V = {Vmn , m ≥ 1, n ≥ 1} and V = {Vmn , m ≥ 1, n ≥ 1} be two double arrays of independent random elements in a real separable Banach space such that V and V are independent copies of each other Then Smn → a.s as m ∨ n → ∞ m α nβ (3.5) if and only if m i=1 n j=1 (Vij mα nβ − Vij ) → a.s as m ∨ n → ∞ and Smn P → as m ∨ n → ∞ mα nβ (3.6) Proof The implication (3.5)⇒(3.6) is obvious To prove the implication (3.6)⇒(3.5), set m n Vij , m ≥ 1, n ≥ Smn = i=1 j=1 Then for all m ≥ 1, n ≥ 1, Smn /(mα nβ ) and Smn /(mα nβ ) are i.i.d real valued random variables Let µmn denote a median of Smn /(mα nβ ), m ≥ 1, n ≥ By the second half of (3.6), µmn → as m ∨ n → ∞ (3.7) By the strong symmetrization inequality (see, e.g., Gut [12, p 134]), we have for all ε > P Smn /(mα nβ ) − µmn > ε sup k≤m∨n≤l ≤ 2P sup k≤m∨n≤l ≤ 2P sup m∨n≥k Smn /(mα nβ ) − Smn /(mα nβ ) >ε (3.8) Smn − Smn mα nβ >ε → as k → ∞ (by the first half of (3.6)) Letting l → ∞ in (3.8), we have P supm∨n≥k Smn /(mα nβ ) − µmn > ε → as k → ∞ This means that Smn − µmn → a.s as m ∨ n → ∞ (3.9) m α nβ By combining (3.7) and (3.9), we obtain (3.5) The second lemma shows that if Vmn ≤ mα nβ a.s., m ≥ 1, n ≥ then Smn /(mα nβ ) obeying the WLLN as m ∨ n → ∞ is indeed equivalent to its convergence in Lp to as m ∨ n → ∞ for any p > Lemma 3.3 Let α > 0, β > and let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent symmetric random elements in a real separable Banach space such that Vmn ≤ mα nβ a.s for all m ≥ 1, n ≥ If Smn P → as m ∨ n → ∞, (3.10) m α nβ then for all p > 0, Smn → in Lp as m ∨ n → ∞ (3.11) m α nβ Proof Let p > and let ε > be arbitrary Let Kmn = max1≤i≤m,1≤j≤n Vij , m ≥ 1, n ≥ Since Vmn ≤ mα nβ a.s for all m ≥ 1, n ≥ 1, Kmn ≤ mα nβ a.s., m ≥ 1, n ≥ (3.12) By (3.10), there exists a positive integer N such that whenever m ∨ n ≥ N , P ( Smn ≥ mα nβ ε) ≤ × 3p (3.13) Now for all A > 0, A A/3 tp−1 P ( Smn > mα nβ t)dt = 3p P ( Smn > 3mα nβ t)dt A/3 ≤ 3p A/3 tp−1 P ( Smn > mα nβ t)dt + p ≤ ≤ 4(3ε) + p 4(3ε)p + p tp−1 P (Kmn > mα nβ t)dt (by Lemma 2.2) A/3 A/3 tp−1 P ( Smn > mα nβ t)dt + 3p ε tp−1 P (Kmn > mα nβ t)dt (by (3.13)) A tp−1 P ( Smn > mα nβ t)dt + 3p tp−1 P (Kmn > mα nβ t)dt (by (3.12)) (3.14) It follows from (3.14) that for all A > A tp−1 P ( Smn > mα nβ t)dt ≤ and hence E Smn m α nβ 8(3ε)p + × 3p p tp−1 P (Kmn > mα nβ t)dt ∞ p tp−1 P ( Smn > mα nβ t)dt =p A tp−1 P ( Smn > mα nβ t)dt = p lim A→∞ (3.15) tp−1 P (Kmn > mα nβ t)dt ≤ 8(3ε)p + (2p)3p Note that for all m ≥ 1, n ≥ 1, Kmn ≤ maxk≤m,l≤n Skl and so by Lemma 2.3 and (3.10) we have for t > P (Kmn > mα nβ t) ≤ P ( max Skl > mα nβ t/4) k≤m,l≤n (3.16) ≤ CP ( Smn > mα nβ t/(4C)) → as m ∨ n → ∞ Hence by the Lebesgue dominated convergence theorem, (3.16) implies that tp−1 P (Kmn > mα nβ t)dt → as m ∨ n → ∞ (3.17) The conclusion (3.11) follows from (3.15), (3.17), and the arbitrariness of ε > We use the L´evy type inequality for double arrays of independent symmetric random elements (Lemma 2.3) as a key tool to prove the following lemma This lemma is a double sum version of Lemma 3.2 of de Ascota [5] Lemma 3.4 Let α > 0, β > and let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent symmetric random elements in a real separable Banach space Then Smn → a.s as m ∨ n → ∞ mα nβ if and only if 2m −1 2n −1 i=2m−1 j=2n−1 mα 2nβ Vij → a.s as m ∨ n → ∞ (3.18) (3.19) Proof Let 2m −1 2n −1 Vij , m ≥ 1, n ≥ Tmn = i=2m−1 j=2n−1 The implication (3.18)=⇒ (3.19) is immediate since for all m ≥ 2, n ≥ Tmn = S(2m − 1; 2n − 1) − S(2m − 1; 2n−1 − 1) − S(2m−1 − 1; 2n − 1) + S(2m−1 − 1; 2n−1 − 1) Next, we assume that (3.19) holds Since the array {Tmn , m ≥ 1, n ≥ 1} is comprised of independent random elements, by the Borel-Cantelli lemma ∞ ∞ P ( Tmn > 2mα 2nβ ε) < ∞ for all ε > m=1 n=1 (3.20) Let u Mrs = v Vij , r ≥ 1, s ≥ max 2r−1 ≤u≤2r −1,2s−1 ≤v≤2s −1 i=2r−1 j=2s−1 By Lemma 2.3 and (3.20), we have ∞ ∞ ∞ ∞ P (Mrs > 2rα 2sβ ε) ≤ C r=1 s=1 P ( Trs > 2rα 2sβ ε/C) < ∞ for all ε > r=1 s=1 This ensures that Mrs → a.s as r ∨ s → ∞ 2rα 2sβ For m ≥ 1, n ≥ 1, let k ≥ 0, l ≥ be such that (3.21) 2k ≤ m ≤ 2k+1 − and 2l ≤ n ≤ 2l+1 − Then for m ≥ 1, n ≥ 1, k+1 l+1 Smn ≤ Mrs r=1 s=1 and so Smn mα nβ k+1 l+1 ≤ r=1 s=1 2rα 2sβ Mrs 2kα 2lβ 2rα 2sβ (3.22) Note that k+1 l+1 sup k≥1,l≥1 r=1 s=1 2rα 2sβ < ∞, and 2kα 2lβ 2rα 2sβ = for every fixed r, s k∨l→∞ 2kα 2lβ lim (3.23) Hence from (3.21) and (3.23), we get by applying Lemma 2.4 that k+1 l+1 r=1 s=1 2rα 2sβ Mrs → a.s as k ∨ l → ∞ 2kα 2lβ 2rα 2sβ (3.24) The conclusion (3.18) follows from (3.22) and (3.24) Proof of Theorem 3.1(i) Assume that (3.2) holds By Lemma 3.2, it is enough to prove the theorem assuming the {Vmn , m ≥ 1, n ≥ 1} are symmetric Set Wmn = Vmn I( Vmn ≤ mα nβ ), m ≥ 1, n ≥ By Markov’s inequality and (3.1) ∞ ∞ ∞ P Vmn > mα nβ ≤ n=1 m=1 ∞ E Vmn p < ∞ mαp nβp n=1 m=1 (3.25) Also by (3.1), ∞ ∞ E Wmn p < ∞ mαp nβp n=1 m=1 (3.26) By (3.25) and the Borel-Cantelli lemma, it suffices to prove m i=1 n j=1 m α nβ Wij → a.s as m ∨ n → ∞ (3.27) Using (3.25) and the Borel-Cantelli lemma again, it follows from (3.2) that m i=1 n j=1 m α nβ Wij P → as m ∨ n → ∞ Thus, Lemma 3.3 ensures that m i=1 n j=1 m α nβ Wij = 2m −1 2n −1 i=1 j=1 2mα 2nβ → in L1 as m ∨ n → ∞ and so 2m −1 2n −1 i=2m−1 j=2n−1 2mα 2nβ Wij − Wij 2m −1 2n−1 −1 Wij i=1 j=1 2β 2mα 2(n−1)β − 2m−1 −1 2n −1 i=1 j=1 Wij α (m−1)α 2 2nβ + 2m−1 −1 2n−1 −1 Wij i=1 j=1 α β (m−1)α (n−1)β 2 2 (3.28) → a.s as m ∨ n → ∞, (3.29) → in L1 as m ∨ n → ∞ Now if we can show that 2m −1 i=2m−1 2n −1 j=2n−1 Wij − E 2m −1 i=2m−1 2n −1 j=2n−1 Wij 2mα 2nβ then it follows from (3.28) that 2m −1 2n −1 i=2m−1 j=2n−1 2mα 2nβ Wij → a.s as m ∨ n → ∞ which yields (3.27) via Lemma 3.4 To prove (3.29), note that ∞ ∞ 2m −1 i=2m−1 E Wij − E 2m −1 i=2m−1 2n −1 j=2n−1 p Wij 2mαp 2nβp m=1 n=1 ∞ 2n −1 j=2n−1 2m −1 i=2m−1 ∞ ≤C m=1 n=1 ∞ ∞ ≤C k=1 l=1 2n −1 j=2n−1 E 2mαp 2nβp Wij p (by (2.2) of Lemma 2.1) (3.30) E Wkl p < ∞ (by (3.26)) k αp lβp By Lemma 2.5, (3.29) follows from (3.30) The proof of Theorem 3.1 (i) is completed The proof of Theorem 3.1 (ii) is similar and we omit the details but we point out that (2.3) of Lemma 2.1 is used instead of (2.2) Corollary 3.5 Let α > 0, β > and let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent random elements in a real separable Banach space (i) Assume that for some ≤ p ≤ 2, some ε > 0, and all m ≥ 1, n ≥ that E Vmn p ≤C mαp−1 nβp−1 ((log(m + 1))(log(n + 1)))1+ε (3.31) Then (3.2) and (3.3) are equivalent (ii) Assume that for some p > 1, some ε > and all m ≥ 1, n ≥ that E Vmn 2p ≤C mp(2α−1) np(2β−1) ((log(m + 1))(log(n + 1)))1+ε (3.32) Then (3.2) and (3.3) are equivalent Proof (i) Note that by (3.31) ∞ ∞ ∞ ∞ E Vmn p ≤C p−1 , the condition (3.31) is automatic (ii) If p > and α ∧ β > 1/2, the condition (3.32) is automatic By the same method that is used in the proof of Theorem 3.1, we obtain in Theorem 3.7 a Marcinkiewicz-Zygmund type SLLN for double arrays of i.i.d random elements in arbitrary real separable Banach spaces We also omit the details Theorem 3.7 was originally proved by Mikosch and Norvaiˇsa [13, Corollary 4.2] and by Giang [14, Theorem 1.1] using a different method Theorem 3.7 Let ≤ p < and let {Vmn , m ≥ 1, n ≥ 1} be a double array of i.i.d random elements in a real separable Banach space with E( V11 p log+ V11 ) < ∞ Then Smn P → as m ∨ n → ∞ (mn)1/p (3.33) Smn → a.s as m ∨ n → ∞ (mn)1/p (3.34) if and only if Remark 3.8 In the one dimensional case, for i.i.d random elements {Vn , n ≥ 1}, de Acosta [5, Theorem 3.1] showed that under the condition E V1 p < ∞ where ≤ p < 2, the WLLN implies the SLLN with norming sequence {n1/p , n ≥ 1} This is no longer valid in the multidimensional case To see this, consider a double array of i.i.d symmetric real valued random variables {Xmn , m ≥ 1, n ≥ 1} with E|X11 |p < ∞ and E(|X11 |p log+ |X11 |) = ∞ for some ≤ p < m n Let Smn = i=1 j=1 Xij , m ≥ 1, n ≥ Then by Theorem 3.2 of Rosalsky and Thanh [15], we obtain the WLLN (3.33) However, by Theorem 3.2 of Gut [16], the corresponding SLLN (3.34) does not hold Applications In this section, we will apply the main results to obtain SLLNs for double arrays of independent random elements in a real separable Rademacher type p (1 ≤ p ≤ 2) Banach space The following theorem, which is a Kolmogorov type SLLN, is a part of Theorem 3.1 of Rosalsky and Thanh [10] (see also Thanh [17, Theorem 2.1] for the real valued random variables case) However, the proof we present here is entirely different Theorem 4.1 Let ≤ p ≤ and let X be a real separable Rademacher type p Banach space Let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent mean random elements in X If ∞ ∞ E||Vmn ||p < ∞, mαp nβp m=1 n=1 (4.1) Smn = a.s mα nβ (4.2) where α > 0, β > 0, then the SLLN lim m∨n→∞ obtains Proof By Theorem 3.1 (i), it suffices to show that Smn P − → as m ∨ n → ∞ mα nβ (4.3) Since X is of Rademacher type p, it follows from (2.1) that E Smn mα nβ p ≤ C αp m nβp m n E Vij p → as m ∨ n → ∞ i=1 j=1 by (4.1) and the Kronecker lemma for double series (see, e.g., M´oricz [18, Theorem 1]) noting that the summands in (4.1) are nonnegative Hence (4.3) follows The proof is completed The following two theorems can be proved by the same method We omit the details Theorem 4.2 and Theorem 4.5 are, respectively, a Brunk-Chung type and a Marcinkiewicz-Zygmund type SLLN for double arrays of independent random elements in Rademacher type p Banach spaces Theorem 4.5 was originally obtained by Giang [14, Theorem 1.2] using a different method of proof Theorem 4.5 will follow immediately from Corollary 3.2 of Rosalsky and Thanh [10] if hypothesis that X is of Rademacher type p is strengthened to X being of Rademacher type q for some q ∈ (p, 2] Theorem 4.2 Let q ≥ 1, ≤ p ≤ and let X be a real separable Rademacher type p Banach space Let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent mean random elements in X If ∞ ∞ E||Vmn ||pq < ∞, mαpq−q+1 nβpq−q+1 m=1 n=1 where α > 0, β > 0, then the SLLN (4.2) obtains 10 Corollary 4.3 Let α > 0, β > 0, ≤ p ≤ and let X be a real separable Rademacher type p Banach space Let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent mean random elements in X Assume that for some q ≥ 1, some ε > 0, and all m ≥ 1, n ≥ that pq E Vmn (mαp−1 nβp−1 )q ≤C (log(m + 1))(log(n + 1)) 1+ε (4.4) Then the SLLN (4.2) obtains Proof The proof is similar to that of Corollary 3.5 Remark 4.4 Suppose that ≤ p ≤ and q ≥ are such that supm≥1,n≥1 E Vmn α ∧ β > p−1 , then the condition (4.4) is automatic pq < ∞ If Theorem 4.5 Let ≤ p < and let X be a real separable Rademacher type p Banach space Let {Vmn , m ≥ 1, n ≥ 1} be a double array of i.i.d mean random elements in X If E( V11 p log+ V11 ) < ∞, then the SLLN Smn = a.s lim m∨n→∞ (mn)1/p obtains An Interesting Example The following example of Rosalsky and Thanh [10, Example 4.1] demonstrates that Theorem 4.1 can fail for < p ≤ if X is not of Rademacher type p or if X is of Rademacher type p but the double series of (4.1) diverges Apropos of Theorem 3.1, the example then also shows that (3.2) and (3.3) can both fail when (3.1) holds However, it follows from Theorem 4.1 (and also from Theorem 3.2 of Rosalsky and Thanh [10]) that if (3.1) holds with p = 1, then (3.2) and (3.3) both hold Example 5.1 Let ≤ q < p ≤ and consider the real separable Banach space q consisting of 1/q ∞ absolute q th power summable real sequences v = {vk , k ≥ 1} with norm v = ( k=1 |vk |q ) Let v (k) denote the element of q having in its k th position and elsewhere, k ≥ Let ϕ : N × N → N be a one-to-one and onto mapping Let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent random elements in q by requiring the {Vmn , m ≥ 1, n ≥ 1} to be independent with , m ≥ 1, n ≥ is not of Rademacher type p Note that P Vmn = v (ϕ(m,n)) = P Vmn = −v (ϕ(m,n)) = Let α = β = 1/q It is well known that ∞ ∞ q ∞ ∞ E Vmn p = and so (4.1) holds but (4.2) fails since for all m ≥ 1, n ≥ 1, Now it is also well known that q ∞ Smn (mn)1/q = 1/q 1/q = a.s α β m n m n is of Rademacher type q However, ∞ ∞ ∞ E Vmn q = =∞ αq nβq m mn m=1 n=1 m=1 n=1 and we see from (5.1) that (4.2) fails 11 (5.1) References [1] Rosalsky, A., and Thanh, L.V 2006 Strong and weak laws of large numbers for double sums of independent random elements in Rademacher type p Banach spaces Stochastic Anal Appl 24:1097–1117 [2] Pyke, R 1973 Partial sums of matrix arrays, and Brownian sheets In Stochastic Analysis: A Tribute to the Memory of Rollo Davidson, Kendall, D.G., Harding, E.F., Eds.; John Wiley, London, 331–348 [3] Taylor, R.L 1978 Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces, Lecture Notes in Mathematics Vol 672 Springer-Verlag, Berlin [4] Hoffmann-Jørgensen, J., and Pisier, G 1976 The law of large numbers and the central limit theorem in Banach spaces Ann Probab 4:587–599 [5] de Acosta, A 1981 Inequalities for B-valued random vectors with applications to the strong law of large numbers Ann Probab 9:157–161 [6] Hoffmann-Jørgensen, J 1974 Sums of independent Banach space valued random variables Studia Math 52:159–186 [7] Etemadi, N 1991 Maximal inequalities for partial sums of independent random vectors with multi-dimensional time parameters Comm Statist Theory Methods 20:3909–3923 [8] Lo`eve, M 1977 Probability Theory I, 4th Ed.; Springer-Verlag, New York [9] Stadtmă uller, U., and Thanh, L V 2011 On the strong limit theorems for double arrays of blockwise M -dependent random variables Acta Math Sin (Engl Ser.) 27:1923–1934 [10] Rosalsky, A., and Thanh, L.V 2007 On almost sure and mean convergence of normed double sums of Banach space valued random elements Stochastic Anal Appl 25:895–911 [11] Etemadi, N 1985 Tail probabilities for sums of independent Banach space valued random variables Sankhy¯ a Ser A 47:209–214 [12] Gut, A 2005 Probability: A graduate course Springer-Verlag, New York [13] Mikosch, T., and Norvaiˇsa, R 1987 Strong laws of large numbers for fields of Banach space valued random variables Probab Theory Related Fields 74:241–253 [14] Giang, N.V 1995 Marcinkiewicz-Zygmund laws for Banach space valued random variables with multidimensional parameters Teor Veroyatnost i Primenen 40:213–219 (in Russian); English translation in Theory Probab Appl 40:175–181 [15] Rosalsky, A., and Thanh, L.V 2009 Weak laws of large numbers of double sums of independent random elements in Rademacher type p and stable type p Banach spaces Nonlinear Anal 71:e1065–e1074 [16] Gut, A 1978 Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices Ann Probab 6:469–482 [17] Thanh, L V 2005 Strong law of large numbers and Lp -convergence for double arrays of independent random variables Acta Math Vietnam 30:225–232 12 [18] M´ oricz, F 1981 The Kronecker lemmas for multiple series and some applications Acta Math Acad Sci Hungar 37:39–50 Andrew Rosalsky Department of Statistics University of Florida Gainesville Florida 32611 − 8545, USA E-mail: rosalsky@stat.ufl.edu Le Van Thanh and Nguyen Thi Thuy Department of Mathematics Vinh University 182 Le Duan, Vinh, Nghe An Vietnam Email: levt@vinhuni.edu.vn; thuynt.tc3@nghean.edu.vn 13 ... underlying Banach space In Theorem 3.1, condition (3.1) is a Kolmogorov type condition for the SLLN for double arrays whereas condition (3.4) is a Brunk-Chung type condition for the SLLN for double. .. lemmas The applications of the main results are presented in Section Section contains an example pertaining to Theorems 3.1 and 4.1 Preliminaries In this section, notation, technical definitions,... established Theorem 3.1 considers the independent but nonidentically distributed case while Theorem 3.7 considers the i.i.d case In these theorems, no assumptions are made concerning the geometry of the

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