Research report, "Some convergence theorems for arrays of two average index of random elements in Banach spaces with integrable conditions on" pptx

 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 EV  < ∞ 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 iu m ,y n jv 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 iu m ,y n jv 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 iu m ,y n jv 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 EA 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 EA mnij V ij  = EA mnij V ij I(V ij   a o ) + EA mnij V ij I(V ij  > a o )  a o E|A mnij | + EA mnij V ij I(V ij  > a o ) < ∞ implying that E(A mnij V ij ) exists. Set c mn = 1 sup x m iu m ,y n jv 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 EA mnij (V  mnij − b  mnij ) p  r/p + C u m  i=x m v n  j=y n EA 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 iu m ,y n jv 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 iu m ,y n jv n (E|A mnij |) 1−r  aM sup x m iu m ,y n jv 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 iu m ,y n jv 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