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u m i=x m v n j=y n A mnij V ij {V ij ; i, j ∈ Z} {A mnij ; x m i u m , y n j v n , m 1, n 1} {x m , m ≥ 1} {u m , m ≥ 1} {y n , n ≥ 1} {v n , n ≥ 1} {V ij ; i, j ∈ Z} (Ω, F, P) {A mnij ; x m i u m , y n j v n , m 1, n 1} {x n , n ≥ 1} {u n , n ≥ 1} {y n , n ≥ 1} {v n , n ≥ 1} u n −x n > 0 n ≥ 1 u n −x n → ∞ n → ∞ v n −y n > 0 n ≥ 1 v n − y n → ∞ n → ∞ u m i=x m v n j=y n A mnij V ij L p −→ 0. v n j=u n A nj V j L 1 expected value mean V EV Pettis integral provided it exists V expected value EV ∈ X if f(EV ) = Ef (V ) f ∈ X ∗ X ∗ dual X EV EV < ∞ 1 {A nj ; u n j v n , n ≥ 1} {V j , j ∈ Z} X · (Ω, F, P) {F n , n ≥ 1} σ F n ≥ 1 E F n (Y ) Y F n {V j , j ∈ Z} {A nj } p {F n } > 0 a o = a o () > 0 sup n≥1 v n j=u n |A nj | p E F n (V j p I(V j > a o )) < a.s. A nj = a nj u n j v n , n ≥ 1 F n = {∅, Ω} n ≥ 1 {A nj } p {F n } {|a nj | p } {V j p , j ∈ Z} Lemma 1. {k mn , m ≥ 1, n ≥ 1} lim m∨n→∞ k mn = ∞ {X ij ; i, j ∈ Z} sup a>0 sup n≥1 1 k mn u m i=x m v n i=y n aP{|X ij | > a} M < ∞, (2.1) and lim a→+∞ sup n≥1 1 k mn u m i=x m v n j=y n aP{|X ij | > a} = 0. (2.2) 1 k p mn u m i=x m v n i=y n E(|X ij | p I(|X ij | k mn )) → 0 as m ∨ n → ∞ (p > 1). (2.3) Proof. 1 k p mn u m i=x m v n j=y n E(|X ij | p I(|X ij | k mn )) = = 1 k p mn u m i=x m v n j=y n E(|X ij | p I(|X ij | 1)) + 1 k p mn u m i=x m v n j=y n k mn l=2 E(|X ij | p I(l − 1 < |X ij | l )) =: A mn + B mn . We first verify that lim m∨n→∞ A mn = 0. A mn = 1 k p mn u m i=x m v n i=y n E(|X ij | p I(|X ij | 1)) = 1 k p mn u m i=x m v n j=y n ∞ l=1 E(|X ij | p I( 1 l + 1 < |X ij | 1 l )) 1 k p mn u m i=x m v n j=y n ∞ l=1 1 l p P{ 1 l + 1 < |X ij | 1 l } = 1 k p mn u m i=x m v n j=y n ∞ l=1 1 l p P{|X ij | > 1 l + 1 } − P{|X ij | > 1 l } = 1 k p mn u m i=x m v n j=y n ∞ l=1 1 l p − 1 (l + 1) p P{|X ij | > 1 l + 1 } = 1 k p−1 mn ∞ l=1 1 l p − 1 (l + 1) p (l + 1) 1 k mn u m i=x m v n j=y n 1 l + 1 P{|X ij | > 1 l + 1 } M 1 k p−1 mn ∞ l=1 1 l p − 1 (l + 1) p (l + 1) (by (2.1)) = M 1 k p−1 mn ( ∞ l=1 1 l p + 1) → 0 as m ∨ n → ∞. (2.4) Next, we will show that lim m∨n→∞ B mn = 0. In deed, since k mn l=2 (l p − (l − 1) p ) k p−1 mn (l − 1) = 1 k p−1 mn k mn l=2 l p (l − 1)l + k mn k mn − 1 − 2 p−1 k p−1 mn 2 k p−1 mn k mn l=2 l p−2 + k mn k mn − 1 4, By (2.2) we have B mn = 1 k p mn u m i=x m v n j=y n k mn l=2 E(|X ij | p I(l − 1 < |X ij | l )) 1 k p mn u m i=x m v n j=y n k mn l=2 l p P{l − 1 < |X ij | l } = 1 k p mn u m i=x m v n j=y n k mn l=2 l p [P{|X ij | > l − 1} − P{|X ij | > l}] = 1 k p mn u m i=x m v n j=y n k mn l=2 [l p − (l − 1) p ]P{|X ij | > l − 1} = k mn l=2 (l p − (l − 1) p ) k p−1 mn (l − 1) 1 k mn u m i=x m v n i=y n (l − 1)P{|X ij | > l − 1} 4. 1 k mn u m i=x m v n j=y n (l − 1)P{|X ij | > l − 1} 4. sup m≥1,n≥1 1 k mn u m i=x m v n j=y n (l − 1)P{|X ij | > l − 1} → 0 as l → ∞. (2.5) So the conclusion (2.3) follows from (2.4) and (2.5). Corollary 1. {a mnij ; x m i u m , y n j v n , m ≥ 1, n ≥ 1} u m i=x m v n j=y n |a mnij | M < ∞ and sup x m iu m ,y n jv n |a mnij | → 0 as m ∨ n → ∞. {X ij ; i, j ∈ Z} {|a mnij |} lim a→+∞ sup m≥1,n≥1 u m j=x m u n j=y n |a mnij |E(|X ij |I(|X ij | > a)) = 0. c mn = 1 sup x m iu m ,y n jv n |a mnij | u m i=x m v n j=y n |a mnij | q E(|X ij | q I(|X ij | c mn )) → 0 as m ∨ n → ∞ (q > 1). Proof. Applying Lemma 1 with k mn = [c mn ] + 1 and X ij is replaced by a mnij c mn X ij . {Y n , n ≥ 1} Bernoulli sequence {Y n , n ≥ 1} P{Y 1 = 1} = P{Y 1 = −1} = 1/2 X ∞ = X × X × X × . . . C(X ) = {(v 1 , v 2 , . . .) ∈ X ∞ : ∞ n=1 v n v n converges in probability}. 1 p 2 X Rademacher type p C (0 < C < ∞) E ∞ n=1 Y n v n p C ∞ n=1 v n p for all (v 1 , v 2 , v 3 , . . .) ∈ C(X ). (2.6) φ 1 p 2 Rademacher type p C (0 < C < ∞) E n j=1 V j p C n j=1 V j p (2.7) {V 1 , V 2 , . . . , V n } 1 < p 2 1 r < p L p l p 2∧p p ≥ 1 a, b ∈ R, min{a, b} max{a, b} a ∧ b a∨b C (0 < C < ∞) Theorem 1. 1 r < p 2 {V ij ; i, j ∈ Z} (Ω, F, P) p X {A mnij ; x m i u m , y n j v n , m 1, n 1} u m i=x m v n j=y n E|A mnij | r M < ∞ (3.1) and sup x m iu m ,y n jv n E|A mnij | r → 0 as m ∨ n → ∞. (3.2) {F mn ; m ≥ 1, n ≥ 1} σ F A mnij , x m i u m , y n j v n F mn {V ij ; i, j ∈ Z} {A mnij } r {F mn } lim a→+∞ sup m≥1,n≥1 u m i=x m v n j=y n |A mnij | r E F mn (V ij r I(V ij > a)) = 0 a.s. (3.3) m ≥ 1 n ≥ 1 {A mnij V ij ; x m i u m , y n j v n } A mnij V ij m ≥ 1, n ≥ 1, x m i u m , y n j v n u m i=x m v n j=y n A mnij V ij L r −→ 0 as m ∨ n → ∞. (3.4) Proof. Since (3.3) there exists a o > 0 such that E u m j=x m v n j=y n |A mnij | r E F mn (V ij r I(V ij > a o )) < 1, m ≥ 1, n ≥ 1. Thus EA mnij V ij I(V ij > a o ) < 1 for all x m i u m , y n j v n , m ≥ 1, n ≥ 1. (3.5) For all m ≥ 1, n ≥ 1, x m i u m , y n j v n , (by (3.1) and (3.5) we have EA mnij V ij = EA mnij V ij I(V ij a o ) + EA mnij V ij I(V ij > a o ) a o E|A mnij | + EA mnij V ij I(V ij > a o ) < ∞ implying that E(A mnij V ij ) exists. Set c mn = 1 sup x m iu m ,y n jv n E|A mnij | r , V mnij = V ij I(V ij c mn ), V mnij = V ij I(V ij > c mn ), b mnij = EV mnij , b mnij = EV mnij . Observe that for each i and j, x m i u m , y n j v n , then V ij = (V mnij −b mnij )+ (V mnij − b mnij ). And since A mnij and V ij are independent for each m, n, i, j we have E(A mnij (V ij − b mnij )) = E(A mnij (V mnij − b mnij )) = 0. Hence E u m i=x m v n j=y n A mnij V ij r = E u m i=x m v n j=y n A mnij (V mnij − b mnij ) + u m i=x m v n j=y n A mnij (V mnij − b mnij ) r CE u m i=x m v n j=y n A mnij (V mnij − b mnij ) r + CE u m i=x m v n j=y n A mnij (V mnij − b mnij ) r (by c r -inequality) C E u m i=x m v n j=y n A mnij (V mnij − b mnij ) p r/p + CE u m i=x m v n j=y n A mnij (V mnij − b mnij ) r C u m i=x m v n j=y n EA mnij (V mnij − b mnij ) p r/p + C u m i=x m v n j=y n EA mnij (V mnij − b mnij ) r C u m i=x m v n j=y n E|A mnij | p E(V ij p I(V ij c mn )) r/p + C u m i=x m v n j=y n E(A mnij V ij r I(V ij > c mn )). Now, by (3.3), for arbitrary > 0 there exists a o > 0 such that for all a ≥ a o . We have E sup m≥1,n≥1 u m j=x m v n j=y n |A mnij | r E F mn (V ij r I(V ij > a)) < . (3.6) This implies sup m≥1,n≥1 u m i=x m v n j=y n E|A mnij | r E(V ij r I(V ij > a)) < ∀a ≥ a o . (3.7) Note that (3.6) means {V ij r ; i, j ∈ Z} is {E|A mnij | r }-uniformly integrable, and then by Corollary 1 with q = p/r, X ij = V ij r and a mnij = E|A mnij | r we get u m i=x m v n j=y n |A mnij | p E(V ij p I(V ij c mn )) → 0 as m ∨ n → ∞. On the other hand (3.6) also implies u m i=x m v n j=y n E(A mnij V ij r I(V ij > c mn ) → 0 as m ∨ n → ∞. Thus E u m i=x m v n j=y n A mnij V ij r → 0 as m ∨ n → ∞. The proof is completed. Theorem 2. 0 < r < 1 {V ij ; i, j ∈ Z} {A mnij ; x m i u m , y n j v n , m 1, n 1} u m i=x m v n j=y n (E|A mnij |) r M < ∞ (3.8) and sup x m iu m ,y n jv n E|A mnij | → 0 as m ∨ n → ∞. (3.9) {F mn ; m ≥ 1, n ≥ 1} σ F A mnij , x m i u m , y n j v n F mn {V ij ; i, j ∈ Z} {A mnij } r {F mn } (3.3) u m i=x m v n j=y n A mnij V ij L r −→ 0 as m ∨ n → ∞. (3.10) Proof. By (3.3), for arbitrary > 0 there exists a > 0 such that E sup m≥1,n≥1 u m j=x m v n j=y n |A mnij | r E F mn (V ij r I(V ij > a)) < 2 , this implies u m i=x m v n j=y n E(A mnij V ij r I(V ij > a)) < 2 , m ≥ 1, n ≥ 1. On the other hand, since E u m j=x m v n j=y n A mnij V ij I(V ij a) r 1/r E u m j=x m v n j=y n A mnij V ij I(V ij a) u m j=x m v n j=y n E(A mnij V ij I(V ij a)) a u m j=x m v n j=y n E|A mnij | a u m j=x m v n j=y n (E|A mnij |) r sup x m iu m ,y n jv n (E|A mnij |) 1−r aM sup x m iu m ,y n jv n (E|A mnij |) 1−r → 0 as m ∨ n → ∞, there exists m o , n o such that for all (m ∨ n ) ≥ (m o ∨ n o ), E u m i=x m v n j=y n A mnij V ij I(V ij a) r 2 . (3.11) Hence, E u m i=x m v n j=y n A mnij V ij r = E u m i=x m v n j=y n A mnij V ij I(V ij a) + u m i=x m v n j=y n A mnij V ij I(V ij > a) r E u m i=x m v n j=y n A mnij V ij I(V ij a) r + E u m i=x m v n j=y n A mnij V ij I(V ij > a) r E u m i=x m v n j=y n A mnij V ij I(V ij a) r + u m i=x m v n j=y n E(A mnij V ij r I(V ij > a)) < for all (m ∨ n) ≥ (m o ∨ n o ), which completes the proof. Remark. {A mnij ; x m i u m , y n j v n , m 1, n 1} F mn m ≥ 1 n ≥ 1 F mn = σ(A mnij , x m i u m , y n j v n ) F mn σ {A mnij ; x m i u m , y n j v n , m 1, n 1} m ≥ 1 n ≥ 1 p 37 φ p 21 p 52 u m i=x m v n j=y n A mnij V ij {V ij ; i, j ∈ Z} {A mnij ; x m i u m , y n j v n } {x m , m ≥ 1} {u m , m ≥ 1} {y n , n ≥ 1} {v n , n ≥ 1} 13 th . EV mnij . Observe that for each i and j, x m i u m , y n j v n , then V ij = (V mnij −b mnij )+ (V mnij − b mnij ). And since A mnij and V ij are independent for each m, n, i, j. a o . (3.7) Note that (3.6) means {V ij r ; i, j ∈ Z} is {E|A mnij | r }-uniformly integrable, and then by Corollary 1 with q = p/r, X ij = V ij r and a mnij = E|A mnij | r we get u m i=x m v n j=y n |A mnij | p E(V ij p I(V ij . 0. c mn = 1 sup x m iu m ,y n jv n |a mnij | u m i=x m v n j=y n |a mnij | q E(|X ij | q I(|X ij | c mn )) → 0 as m ∨ n → ∞ (q > 1). Proof. Applying Lemma 1 with k mn = [c mn ] + 1 and X ij is replaced by a mnij c mn X ij . {Y n , n ≥ 1} Bernoulli
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