467–482DOI 10.4134/BKMS.2010.47.3.467 MEAN CONVERGENCE THEOREMS AND WEAK LAWS OF LARGE NUMBERS FOR DOUBLE ARRAYS OF RANDOM ELEMENTS IN BANACH SPACES Le Van Dung and Nguyen Duy Tien Abstr
Trang 1Bull Korean Math Soc 47 (2010), No 3, pp 467–482
DOI 10.4134/BKMS.2010.47.3.467
MEAN CONVERGENCE THEOREMS AND WEAK LAWS OF LARGE NUMBERS FOR DOUBLE ARRAYS OF RANDOM
ELEMENTS IN BANACH SPACES
Le Van Dung and Nguyen Duy Tien
Abstract For a double array of random elements {V mn ; m ≥ 1, n ≥
1} in a real separable Banach space, some mean convergence theorems
and weak laws of large numbers are established For the mean
conver-gence results, conditions are provided under which k − mn1
Pu m
i=1
Pv n
j=1 (V ij − E(V ij |F ij )) → 0 in L r (0 < r < 2) The weak law results provide
con-ditions for k mn −1PT m
i=1
Pτ n
j=1 (V ij − E(V ij |F ij )) → 0 in probability where
{T m ; m ≥ 1} and {τ n ; n ≥ 1} are sequences of positive integer-valued
random variables, {k mn ; m ≥ 1, n ≥ 1} is an array of positive integers.
The sharpness of the results is illustrated by examples.
1 Introduction
The classical notion of uniform integrability of a sequence {X n ; n ≥ 1} of
integrable random variables is defined through the condition
lim
a→∞sup
n≥1 E|X n |I(|X n | > a) = 0.
Landers and Rogge [4] prove that the uniform integrability condition is suffi-cient in order that a sequence of pairwise independent random variables verifies the weak law of large numbers
Chandra [2] obtains the weak law of large numbers under the condition which is weaker than uniform integrability: the condition of Ces`aro uniform
integrability A sequence {X n ; n ≥ 1} of integrable random variables is said to
be Ces`aro uniformly integrable if
lim
a→∞sup
n≥1
1
k n
k n
X
j=1
E|X j |I(|X j | > a) = 0,
where {k n ; n ≥ 1} is a sequence of positive integers such that lim n→∞ k n = ∞.
Received August 8, 2008; Revised August 25, 2009.
2000 Mathematics Subject Classification 60B11, 60B12, 60F15, 60F25, 60G42.
Key words and phrases martingale type p Banach spaces, double arrays of random
ele-ments, weighted double sums, weak laws of large numbers, mean convergence theorem.
c
°2010 The Korean Mathematical Society
467
Trang 2468 LE VAN DUNG AND NGUYEN DUY TIEN
Definition Let {X n ; n ≥ 1} be a sequence of random variables and {a nj ; 1 ≤
j ≤ v n , n ≥ 1} be an array of constants withPv n
j=1 |a nj | ≤ C for all n ∈ N and
some constant C > 0 The sequence {X n ; n ≥ 1} is {a nj }-uniform integrable if
lim
a→∞sup
n≥1
v n
X
j=1
|a nj |E|X j |I(|X j | > a) = 0,
where {v n ; n ≥ 1} is a sequence of positive integers such that lim n→∞ v n = ∞ Under the condition of {a nj }-uniform integrability, Ord´o˜nez Cabrera [5] ob-tains the weak law of large numbers for weighted sums of pairwise indepen-dent random variables; the condition of pairwise independence can be even dropped, at the price of slightly strengthening the conditions on the weights Recently, Thanh [10] obtains the mean convergence theorems for the weighted sums Pk m
i=1
Pl n
j=1 a mnij (X ij − EX ij ) in L p and the weak laws of large num-bers with random indices for the weighted sumsPT m
i=1
Pτ n
j=1 a mnij (X ij −EX ij),
where {X ij , i ≥ 1, j ≥ 1} is an array of random variable, {a mnij ; m, n, i, j ≥ 1} are constants, {T m ; m ≥ 1} and {τ n ; n ≥ 1} are sequences of positive
integer-valued random variables
Sung [9] introduces the concept of Ces`aro type uniform integrability with
exponent r.
Definition Let {X n ; n ≥ 1} be a sequence of random variables and r > 0 The array {X n ; n ≥ 1} is said to be Ces`aro type uniformly integrable with
exponent r if
sup
n≥1
1
k n
v n
X
j=1
E|X j | r < ∞ and lim
a→∞sup
n≥1
1
k n
v n
X
j=1
E|X j | r I(|X j | > a) = 0,
where {k n ; n ≥ 1} and {v n ; n ≥ 1} are two sequences of positive integers such
that limn→∞ k n= limn→∞ v n = ∞.
In this paper, we not only enlarge on some results of Adler et al [1] and Thanh [10] but we also weaken the suppositions and bring more general results
2 Preliminaries
For a, b ∈ R, min {a, b}, max {a, b} and the integer part of a will be denoted, respectively, by a ∧ b, a ∨ b and [a] Throughout this paper, the symbol C will denote a generic constant (0 < C < ∞) which is not necessarily the same one
in each appearance
Technical definitions relevant to the current work will be discussed in this section Scalora [8] introduced the idea of the conditional expectation of a
random element in a Banach space For a random element V and sub σ-algebra G of F, the conditional expectation E(V |G) is defined analogously to
that in the random variable case and enjoys similar properties
Trang 3MEAN CONVERGENCE THEOREMS AND WLLNS FOR DOUBLE ARRAYS 469
A real separable Banach space X is said to be martingale type p (1 ≤ p ≤ 2) if there exists a finite positive constant C such that for all martingales {S n ; n ≥ 1} with values in X ,
sup
n≥1
EkS n k p ≤ C
∞
X
n=1
EkS n − S n−1 k p
It can be shown using classical methods from martingale theory that if X is of martingale type p, then for all 1 ≤ r < ∞ there exists a finite constant C such
that
E sup
n≥1
kS n k r ≤ CE
̰ X
n=1
kS n − S n−1 k p
!r p
Clearly every real separable Banach space is of martingale type 1 and the real line (the same as any Hilbert space) is of martingale type 2 If a real separable
Banach space of martingale type p for some 1 < p ≤ 2, then it is of martingale type r for all r ∈ [1, p).
It follows from the Hoffmann-Jørgensen and Pisier [3] characterization of
Rademacher type p Banach spaces that if a Banach space is of martingale type
p, then it is of Rademacher type p But the notion of martingale type p is
only superficially similar to that of Rademacher type p and has a geometric
characterization in terms of smoothness For proofs and more details, the reader may refer to Pisier [6, 7]
The following lemma is needed to prove our main results
Lemma 2.1 Suppose that the array of random elements {V mn ; m ≥ 1, n ≥ 1}
is Ces`aro type uniform integrability with exponent r, in the sense that
m≥1,n≥1
1
k mn
u m
X
i=1
v n
X
j=1
EkV ij k r ≤ M < ∞ and
a→∞ sup
m≥1,n≥1
1
k mn
u m
X
i=1
v n
X
j=1
EkV ij k r I(kV ij k r > a) = 0, where {k mn ; m ≥ 1, n ≥ 1} is an array of positive integers such that
lim
m∨n→∞ k mn = ∞.
Then
1
k β r
mn
u m
X
i=1
v n
X
j=1
EkV ij k β I(kV ij k r ≤ k mn ) → 0 as m ∨ n → ∞ if r < β.
Proof.
1
k β r
mn
u m
X
i=1
v n
X
j=1
EkV ij k β I(kV ij k r ≤ k mn)
Trang 4470 LE VAN DUNG AND NGUYEN DUY TIEN
k β r
mn
u m
X
i=1
v n
X
j=1
kXmn
l=1
EkV ij k β I(l − 1 < kV ij k r ≤ l)
k β r
mn
u m
X
i=1
v n
X
j=1
kXmn
l=1
l β−r r EkV ij k r I(l − 1 < kV ij k r ≤ l)
k β r
mn
u m
X
i=1
v n
X
j=1
kXmn
l=1
l β−r r (EkV ij k r I(kV ij k r > l − 1) − EkV ij k r I(kV ij k r > l))
k β r
mn
u m
X
i=1
v n
X
j=1
EkV ij k r I(kV ij k r > 0)
+ 1
k β r
mn
u m
X
i=1
v n
X
j=1
k mnX−1 l=1
³
(l + 1) β−r r − (l) β−r r
´
EkV ij k r I(kV ij k r > l)
=: A mn + B mn
For A mn we have
A mn ≤ 1
k β r −1
mn
sup
m≥1,n≥1
1
k mn
u m
X
i=1
v n
X
j=1
EkV ij k r → 0 as m ∨ n → ∞,
by β
r − 1 > 0 and (2.2) Next, we will show that
lim
m∨n→∞ B mn = 0.
For arbitrary ε > 0, since (2.3) there exists a0 such that
sup
m≥1,n≥1
1
k mn
u m
X
i=1
v n
X
j=1
EkV ij k r I(kV ij k r > a0) < ε
2.
Now for all large enough m ∨ n, we have k mn ≥ ([a0] + 1)
µ
2M
ε
¶ r β−r
and
B mn
k β r
mn
u m
X
i=1
v n
X
j=1
[a0 ]
X
l=1
³
(l + 1) β−r r − (l) β−r r
´
EkV ij k r I(kV ij k r > l)
+ 1
k β r
mn
u m
X
i=1
v n
X
j=1
k mnX−1 l=[a0 ]+1
³
(l + 1) β−r r − (l) β−r r
´
EkV ij k r I(kV ij k r > l)
k β r
mn
u m
X
i=1
v n
X
j=1
[a0]
X
l=1
³
(l + 1) β−r r − (l) β−r r
´
EkV ij k r I(kV ij k r > 1)
Trang 5MEAN CONVERGENCE THEOREMS AND WLLNS FOR DOUBLE ARRAYS 471
+ 1
k β r
mn
u m
X
i=1
v n
X
j=1
k mnX−1 l=[a0 ]+1
³
(l + 1) β−r r − (l) β−r r
´
EkV ij k r I(kV ij k r > [a0] + 1)
k β r
mn
u m
X
i=1
v n
X
j=1
³
([a0] + 1)β−r r − 1´EkV ij k r I(kV ij k r > 1)
+ 1
k β r
mn
u m
X
i=1
v n
X
j=1
³
(k mn)β−r r − ([a0] + 1)β−r r
´
EkV ij k r I(kV ij k r > [a0] + 1)
≤
µ
[a0] + 1
k mn
¶β−r
r
sup
m≥1,n≥1
1
k mn
u m
X
i=1
v n
X
j=1
EkV ij k r
+ sup
m≥1,n≥1
1
k mn
u m
X
i=1
v n
X
j=1
EkV ij k r I(kV ij k r > [a0] + 1) < ε
2 +
ε
2 = ε.
Thus B mn → 0 as m ∨ n → ∞ The proof is completed. ¤
To prove Lemma 2.3, we need the following lemma:
Lemma 2.2 If {X kl , F l ; l ≥ 1}, k = 1, 2, , m are nonnegative
submartin-gales, then {max 1≤k≤m X kl , F l ; l ≥ 1} is a nonnegative submartingale.
Proof For L > l ≥ 1,
E( max
1≤k≤m X kL |F l ) ≥ max
1≤k≤m E(X kL |F l ) ≥ max
1≤k≤m X kl
¤
Let F kl be the σ-field generated by the family of random elements {V ij ; i <
k or j < l}, F 1,1 = {∅; Ω} We have the following lemma:
Lemma 2.3 Let {V ij ; 1 ≤ i ≤ m, 1 ≤ j ≤ n} be a collection of mn random
elements in a martingale type p (1 ≤ p ≤ 2) Banach space with E(V ij |F ij) = 0
for all i, j (1 ≤ i ≤ m, 1 ≤ j ≤ n) Then
°
°
°
m
X
i=1
n
X
j=1
V ij
°
°
°
p
≤ C
m
X
i=1
n
X
j=1
EkV ij k p;
1≤k≤mmax
1≤l≤n
°
°
°
k
X
i=1
l
X
j=1
V ij
°
°
°> ε
≤
C
ε p
m
X
i=1
n
X
j=1
EkV ij k p ∀ε > 0.
Proof First, we prove (2.3) Set S kn =Pk i=1Pn j=1 V ij , since E(V ij |F ij) = 0
we have that {S kn , G k = F k+1,1 ; 1 ≤ k ≤ m} is a martingale Thus
E
°
°
°
m
X
i=1
n
X
j=1
V ij
°
°
°
p
≤ E max 1≤k≤m kS kn k p ≤ C
m
X
k=1
Ek
n
X
j=1
V kj k p
(2.5)
Trang 6472 LE VAN DUNG AND NGUYEN DUY TIEN
On the other hand, for each k (1 ≤ k ≤ m), {Pl j=1 V kj , G kl = F k,l+1 ; 1 ≤ l ≤ n}
is a martingale Hence,
Ek
n
X
j=1
V kj k p ≤ E max
1≤l≤n k
l
X
j=1
V kj k p ≤ C
n
X
l=1
EkV kl k p
(2.6)
Combining (2.5) and (2.6) yields the conclusion (2.3).
Next, we will prove (2.4) Set S kl=Pk i=1Pl j=1 V ij , Y l= max1≤k≤m kS kl k.
If σ l is a σ-field generated by {V ij ; 1 ≤ i ≤ m, 1 ≤ j ≤ l}, then for each
l (1 ≤ l ≤ n), σ l ⊂ F i,l+1 for all i ≥ 1, which follows that E(V i,l+1 |σ l) =
E(E(V i,l+1 |F i,l+1 )|σ l) = 0 Thus, we have
E(S k,l+1 |σ l ) = E(S kl |σ l) +
k
X
i=1
E(V i,l+1 |σ l ) = S kl
It means that {S kl , σ l ; 1 ≤ l ≤ n} is a martingale Hence, {kS kl k, σ l ; 1 ≤ l ≤
n} is a nonnegative submartingale for each k = 1, 2, , m, which follows by
Lemma 2.1 that {Y l , σ l ; 1 ≤ l ≤ n} is a nonnegative submartingale Applying
Kolmogorov’s inequality, we obtain
P
1≤k≤mmax
1≤l≤n
°
°
°
k
X
i=1
l
X
j=1
V ij
°
°
°> ε
= P { max 1≤l≤n Y l > ε} ≤ 1
ε p EY n p
= 1
ε p E max 1≤k≤m kS kn k p ≤ C
ε p
m
X
k=1
Ek
n
X
j=1
V kj k p
ε p
m
X
k=1
E max 1≤l≤n k
l
X
j=1
V kj k p
ε p
m
X
k=1
n
X
l=1
EkV kl k p
= C
ε p
m
X
i=1
n
X
j=1
EkV ij k p
3 Main results With the preliminaries accounted for, the main results may now be
estab-lished In the following we let {V mn ; m ≥ 1, n ≥ 1} be an array of random elements defined on a probability (Ω, F, P ) and taking values in a real sepa-rable Banach space X with norm k · k, F kl be a σ-field generated by {V ij ; i <
k or j < l}, F 1,1 = {∅; Ω} Let {u n ; n ≥ 1}, {v n ; n ≥ 1} be sequences of
posi-tive integers such that limn→∞ u n= limn→∞ v n = ∞, {k mn ; m ≥ 1, n ≥ 1} be
an array of positive integers such that limm∨n→∞ k mn = ∞.
Trang 7MEAN CONVERGENCE THEOREMS AND WLLNS FOR DOUBLE ARRAYS 473
Theorem 3.1 Let 1 ≤ r < p ≤ 2, Banach space X be a martingale type p.
Suppose that {V mn ; m ≥ 1, n ≥ 1} satisfies the Ces`aro type uniform integrability
with exponent r, in the sense that (2.1) and (2.2) hold Then
1
k mn1
u m
X
i=1
v n
X
j=1 (V ij − E(V ij |F ij )) → 0 in L r as n ∨ m → ∞.
Proof For each m, n, 1 ≤ i ≤ u m , 1 ≤ j ≤ v n, set
V 0
mnij = V ij I(kV ij k r ≤ k mn ), V 00
mnij = V ij I(kV ij k r > k mn ),
U 0 mnij = E(V 0
mnij |F ij ), U 00
mnij = E(V 00
mnij |F ij ).
Observe that for each i and j, 1 ≤ i ≤ u m , 1 ≤ j ≤ v n, we have
V ij − E(V ij |F ij ) = (V mnij 0 − U mnij 0 ) + (V mnij 00 − U mnij 00 )
and E(V 0
mnij − U 0
mnij |F ij ) = E(V 00
mnij − U 00
mnij |F ij) = 0 Hence,
E
°
°
° 1
k mn1
u m
X
i=1
v n
X
j=1
V ij
°
°
°
r
k mn E
°
°
°
u m
X
i=1
v n
X
j=1
(V 0 mnij − U 0
mnij) +
u m
X
i=1
v n
X
j=1
(V 00 mnij − U 00
mnij)
°
°
°
r
k mn E
°
°
°
u m
X
i=1
v n
X
j=1 (V mnij 0 − U mnij 0 )
°
°
°
r
+ C 1
k mn
E
°
°
°
u m
X
i=1
v n
X
j=1
(V 00 mnij − U 00
mnij)
°
°
°
r (by c r-inequality)
k mn
E
°
°
°
u m
X
i=1
v n
X
j=1
(V 0 mnij − U 0
mnij)
°
°
°
p
r/p
+ C 1
k mn E
°
°
°
u m
X
i=1
v n
X
j=1 (V mnij 00 − U mnij 00 )
°
°
°
r
(by Liapunov’s inequality)
k mn
u m
X
i=1
v n
X
j=1
Ek(V mnij 0 − U mnij 0 )k p
r/p
+ C 1
k mn
u m
X
i=1
v n
X
j=1
Ek(V mnij 00 − U mnij 00 )k r (by Lemma 2.3)
Trang 8474 LE VAN DUNG AND NGUYEN DUY TIEN
k mn
u m
X
i=1
v n
X
j=1
EkV ij k p I(kV ij k r ≤ k mn)
r/p
+ C 1
k mn
u m
X
i=1
v n
X
j=1
EkV ij k r I(kV ij k r > k mn) (by c r-inequality)
≤ C
1
k p r
mn
u m
X
i=1
v n
X
j=1
EkV ij k p I(kV ij k r ≤ k mn)
r/p
+ C
sup
m≥1,n≥1
1
k mn
u m
X
i=1
v n
X
j=1
EkV ij k r I(kV ij k r > k mn)
→ 0 as m ∨ n → ∞,
by Lemma 2.1 with β = p and (2.2) The proof is completed. ¤
Corollary 3.2 Let 1 ≤ r < p ≤ 2, Banach space X be a martingale type p.
Let {A mnij ; m ≥ 1, n ≥ 1, 1 ≤ i ≤ u m , 1 ≤ j ≤ v n } be an array of random variables satisfying Pu m
i=1
Pv n
j=1 E|A mnij | r ≤ C < ∞ for all m ≥ 1, n ≥ 1 Suppose that {A mnij ; 1 ≤ i ≤ u m , 1 ≤ j ≤ v n } and {V ij ; i ≥ 1, j ≥ 1} are
independent for all m ≥ 1, n ≥ 1 Assume that the following conditions hold:
(i) {kV mn k r ; m ≥ 1, n ≥ 1} is {E|A mnij | r }-uniformly integrable, in the sense that
lim
a→∞ sup
m≥1,n≥1
u m
X
i=1
v n
X
j=1
EkA mnij V ij k r I(kV ij k > a) = 0;
(ii) sup
1≤i≤u m ,1≤j≤v n
E|A mnij | → 0 as m ∨ n → ∞.
Then
u m
X
i=1
v n
X
j=1
A mnij (V ij − E(V ij |F ij )) → 0 in L r as m ∨ n → ∞.
Proof Let
k mn=
sup
1≤i≤u m ,1≤j≤v n
E|A mnij |
Then k mn → ∞ as m ∨ n → ∞ It is easy to prove that
sup
m≥1,n≥1
u m
X
i=1
v n
X
j=1
EkA mnij V ij k r < ∞.
Hence, take k mn 1/r A mnij V ij instead V ij in Theorem 3.1, we have by (i) that
sup
n≥1,m≥1
1
k mn
u m
X
i=1
v n
X
j=1
Ekk 1/r mn A mnij V ij k r= sup
m≥1,n≥1
u m
X
i=1
v n
X
j=1
EkA mnij V ij k r < ∞.
Trang 9MEAN CONVERGENCE THEOREMS AND WLLNS FOR DOUBLE ARRAYS 475
On the other hand, for all large enough m ∨ n we have
sup
1≤i≤u m ,1≤j≤v n
E|A mnij | ≤ 1,
which follows that
k mn E|A mnij | r ≤ E|A mnij |
r
sup
1≤i≤u m ,1≤j≤v n
E|A mnij | ≤
E|A mnij | r
sup
1≤i≤u m ,1≤j≤v n
E|A mnij | r ≤ 1.
Therefore, for all large enough m ∨ n, k mn |A mnij | r ≤ 1 a.s Thus, it follows
by (i) that
lim
a→∞ sup
m≥1,n≥1
1
k mn
u m
X
i=1
v n
X
j=1
Ekk 1/r
mn A mnij V ij k r I(kV ij k r > a)
≤ lim
a→∞ sup
m≥1,n≥1
u m
X
i=1
v n
X
j=1
EkA mnij V ij k r I(kV ij k > a) = 0.
This implies
lim
a→∞ sup
m≥1,n≥1
1
k mn
u m
X
i=1
v n
X
j=1
Ekk 1/r
mn A mnij V ij k r I(kV ij k r > a) = 0.
Theorem 3.3 Let 0 < r < 1 Suppose that {V mn ; m ≥ 1, n ≥ 1} satisfies the
Ces`aro type uniform integrability with exponent r, in the sense that (2.1) and
(2.2) hold Then
1
k mn1
u m
X
i=1
v n
X
j=1
V ij → 0 in L r as n ∨ m → ∞.
Proof.
E
°
°
° 1
k mn1
u m
X
i=1
v n
X
j=1
V ij
°
°
°
r
= 1
k mn E
°
°
°
u m
X
i=1
v n
X
j=1
V ij
°
°
°
r
k mn
E
°
°
°
u m
X
i=1
v n
X
j=1
V ij I(kV ij k r ≤ k mn) +
u m
X
i=1
v n
X
j=1
V ij I(kV ij k > k mn)
°
°
°
r
k mn E
°
°
°
u m
X
i=1
v n
X
j=1
V ij I(kV ij k r ≤ k mn)
°
°
°
r
+ 1
k mn E
°
°
°
u m
X
i=1
v n
X
j=1
V ij I(kV ij k r > k mn)
°
°
°
r
k mn
E
°
°
°
u m
X
i=1
v n
X
j=1
V ij I(kV ij k r ≤ k mn)
°
°
°
r
Trang 10476 LE VAN DUNG AND NGUYEN DUY TIEN
+ 1
k mn E
°
°
°
u m
X
i=1
v n
X
j=1
V ij I(kV ij k r > k mn)
°
°
°
r
(by Liapunov’s inequality)
≤
1
k mn1
u m
X
i=1
v n
X
j=1
EkV ij kI(kV ij k r ≤ k mn)
r
+
sup
m≥1,n≥1
1
k mn
u m
X
i=1
v n
X
j=1
E kV ij k r I(kV ij k r > k mn)
→ 0 as m ∨ n → ∞,
by Lemma 2.1 with β = 1 and (2.2) The proof is completed. ¤
Corollary 3.4 Let 0 < r < 1 and {A mnij ; m ≥ 1, n ≥ 1, 1 ≤ i ≤ u m , 1 ≤ j ≤
v n } be an array of random variables satisfyingPu m
i=1
Pv n
j=1 E|A mnij | r ≤ C < ∞ for all m ≥ 1, n ≥ 1 Suppose that A mnij and V ij are independent for all
m ≥ 1, n ≥ 1, 1 ≤ i ≤ u m , 1 ≤ j ≤ v n Assume that the following conditions hold:
(i) {kV mn k r ; m ≥ 1, n ≥ 1} is {E|A mnij | r }-uniformly integrable;
(ii) sup
1≤i≤u m ,1≤j≤v n
E|A mnij | → 0 as m ∨ n → ∞.
Then
u m
X
i=1
v n
X
j=1
A mnij V ij → 0 in L r as m ∨ n → ∞.
In the following theorem, we establish the weak law of large numbers with random indices for weighted double sums of random elements
Theorem 3.5 Let 1 ≤ r < p ≤ 2, Banach space X be a martingale type p.
Suppose that {V mn ; m ≥ 1, n ≥ 1} satisfies the Ces`aro type uniform integrability
with exponent r, in the sense that (2.1) and (2.2) hold Let {T n ; n ≥ 1} and
{τ n ; n ≥ 1} be sequences of positive integer-valued random variables such that
n→∞ P {T n > u n } = lim
n→∞ P {τ n > v n } = 0.
Then
k mn1
T m
X
i=1
τ n
X
j=1 (V ij − E(V ij |F ij))−→ 0 as n ∧ m → ∞ P
Proof For arbitrary ε > 0,
(3.3) P
1
k mn1
°
°
°
T m
X
i=1
τ n
X
j=1 (V ij − E(V ij |F ij))
°
°
°> ε