Fuzzy Sets and Systems 159 (2008) 3360 – 3368
www.elsevier.com/locate/fss
Strong laws of large numbers for arrays of rowwise independent
random compact sets and fuzzy random sets夡
Ke-Ang Fu∗ , Li-Xin Zhang
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Received 4 November 2005; received in revised form 3 March 2008; accepted 7 June 2008
Available online 26 June 2008
Abstract
In this paper we obtain some strong laws of large numbers (SLLNs) for arrays of rowwise independent (not necessary identically
distributed) random compact sets and fuzzy random sets whose underlying spaces are separable Banach spaces.
© 2008 Elsevier B.V. All rights reserved.
MSC: 60F15; 60B11
Keywords: Strong laws of large numbers (SLLNs); Random compact set; Fuzzy random set; Stochastically domination; Compact uniform
integrability (CUI)
1. Introduction
In recent years, the theory of random sets and fuzzy random sets has been extensively studied and applied in the
areas of information science, probability and statistics. The general idea of random set has been in existence for some
time. Robbins [24,25] appeared to be the first to provide the concept of random sets, and his early works investigated
the relationships between random sets and geometric probabilities. Later, Kendall [11] and Matheron [19] provided
a comprehensive mathematical theory of random sets which was greatly influenced by the geometric probability
prospective. Their proposed framework exerted a strong influence on the limit theorems developed in recent decades. It
is well-known that strong laws of large numbers (SLLNs) play an important role in probability and statistics, especially
in probability limit theorems. With the development of the theory and application of random sets (cf. [2,7,18,19]),
several variants of SLLNs were built. Among them, Artstein and Vitale [1] proved limit theorems concerning random
sets in and d , and Puri and Ralescu [21] were the first to obtain the SLLNs for independent identically distributed
(i.i.d.) Banach space-valued compact convex random sets. Among others, SLLNs were obtained under more relaxed
conditions, and a detailed survey of these results is available in Taylor and Inoue [26].
The theory of fuzzy sets was introduced by Zadeh [30], and the concept of fuzzy random variables was promoted
by Kwakernaak [16] where useful basic properties were developed. Puri and Ralescu [22] used the concept of fuzzy
random variables to generalize the results of random sets to fuzzy random sets. With respect to laws of large numbers,
Kruse [15] proved an SLLN for i.i.d. fuzzy random variables, and then gave a consistent estimator for the expectation
夡 Project
supported by National Natural Science Foundation of China (nos. 10671176 and 10771192).
∗ Corresponding author. Tel.: +86 13777398392.
E-mail addresses: fukeang@hotmail.com (K.-A. Fu), stazlx@zju.edu.cn (L.-X. Zhang).
0165-0114/$ - see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.fss.2008.06.012
K.-A. Fu, L.-X. Zhang / Fuzzy Sets and Systems 159 (2008) 3360 – 3368
3361
of a fuzzy random variable as an application of the SLLN. Klement et al. [12] considered fuzzy versions of random sets
in Euclidean spaces and obtained an i.i.d. SLLN. Inoue [9] obtained SLLNs for independent, tight fuzzy random sets
and i.i.d. fuzzy random sets in a separable Banach space. Recently, SLLNs have been studied under various conditions,
and one can refer to the following papers [4,6,10,13,17,28,29], and references therein. Also for more detailed results
about limit theorems of random sets and fuzzy random sets, one can refer to Li et al. [18].
However, to the best of our knowledge, most of the authors considered the sequence cases, and limit theorems
concerning arrays of random compact sets or fuzzy random sets are less widely known except Taylor et al. [27] and
Krätschmer [14] where the weak laws of large numbers (WLLNs) and the SLLNs in Euclidean spaces were studied,
respectively. In this paper, we shall study the SLLNs for arrays of rowwise independent random compact sets and
fuzzy random sets whose underlying spaces are separable Banach spaces. The layout of the paper is as follows. The
basic definitions and properties on random compact sets and random fuzzy sets and several useful lemmas are listed in
Section 2. In Section 3 we exhibit some SLLNs and then give the proofs.
2. Properties and lemmas
Throughout this paper, let S be a real separable Banach space with the norm · and the dual space S∗ . For each A ⊂ S,
cl A and co A denote the norm-closure and the closed convex hull of A, respectively. Let C(S) (resp. Cc (S)) denote the
collections of all non-empty compact (resp. non-empty convex compact) subsets of S. Define the Minkowski’s addition
and scalar multiplication, respectively, in C(S) (or Cc (S)) by
A + B = {a + b|a ∈ A, b ∈ B},
A = {a|a ∈ A},
where A, B ∈ C(S) (or Cc (S)) and is a real number. Note that neither C(S) nor Cc (S) are linear spaces even when
S = , one-dimensional Euclidean space. And for A, B ∈ C(S), the Hausdorff distance d H (A, B) of A and B and the
norm A of A are defined by
d H (A, B) = max sup inf a − b, sup inf a − b ,
a∈A b∈B
b∈B a∈A
A = d H (A, {0}) = sup a.
a∈A
Let (, F, P) denote a probability measure space. A random (compact) set is a Borel measurable function F : →
C(S), that is, F −1 (B) = { ∈ ; F() ∩ B = ∅} ∈ F for each B ∈ C(S) (cf. [7,18]). For a random set F in C(S),
there exists a corresponding set co F in Cc (S), which can be used in defining an expected value. A measurable function
f : → S is called a measurable selection of F if f () ∈ F() for every ∈ . Denote by
S F = { f ∈ L 1 (, S); f () ∈ F(), a.e.},
where L 1 (, S) denotes the space of measurable functions f : → S such that n f () dP < ∞. S F = ∅ if and
only if the random variable F() is integrable. The random set F is called integrably bounded if the real-valued
random variable F() is integrable (c.f. [7,18]). Hiai and Umegaki [7] showed that a random set F is integrably
bounded if and only if S F is bounded in L 1 (, S). Thus an integrably bounded random set may take unbounded sets.
For each random set F, the expectation of F, denoted by E(F), is defined by
(2.1)
F dP =
f dP; f ∈ S F ,
E(F) =
where f dP is the usual Bochner integral in L 1 (, S). Define A F dP = { A f dP; f ∈ S F } for A ∈ F. This
definition was introduced by Aumann in 1965 as a natural generalization of the integral of real-valued random
variables
in [2]. For the random set F, if E co F < ∞, then a Bochner integral can be defined as E(co F) = co F dP and
E(co F) ∈ Cc (S) [5].
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K.-A. Fu, L.-X. Zhang / Fuzzy Sets and Systems 159 (2008) 3360 – 3368
For A, B, C and D ∈ C(S), ∈ and X, Y are random sets, immediate properties of the Hausdorff distance lead to
the following:
d H (A, B)||d H (A, B),
d H (A, B) d H (A, C) + d H (C, B),
d H (A + C, B + D) d H (A, B) + d H (C, D),
d H (co A, co B)d H (A, B),
d H (E co X, E co Y ) Ed H (X, Y ).
Also for A1 ⊂ A ⊂ A2 ∈ C(S), the properties aforementioned yield that
d H (A1 , A) d H (A1 , A2 ).
(2.2)
A finite set of random sets {X 1 , . . . , X n } in C(S) is said to be independent if P(X 1 ∈ B1 , . . . , X n ∈ Bn ) = P(X 1 ∈
B1 ) · · · P(X n ∈ Bn ) for every B1 , . . . , Bn ∈ C(S). A family of random set in C(S) is said to be independent if every
finite subset is independent. From the definitions above, we know that most stochastic properties of random sets are
extended from real-valued random variables directly.
Now we begin to state some properties about fuzzy sets. A fuzzy set in S is a function u : S → [0, 1]. Let F(S)
denote the family of the fuzzy subset u satisfying the following conditions:
(a) u is upper semicontinuous, that is, the -level set of u, i.e. u = {x ∈ S; u(x) } is a closed subset of S for each
∈ (0, 1],
(b) {x ∈ S : u(x) > 0} has compact closure,
(c) {x ∈ S : u(x) = 1} = ∅.
For u ∈ F(S), the support of u is defined as supp u = {x ∈ S : u(x) > 0}, and the assumption of compact closure
implies that support of u is norm bounded. A linear structure in F(S) is defined by the operations as follows:
(u + v)(x) = sup min[u(y), v(z)],
y+z=x
(u)(x)
u(−1 x) if = 0;
if = 0,
I0 (x)
where u, v ∈ F(S), ∈ , and I0 (·) is an indicator function. The linear structure aforementioned implies the following
properties:
(u + v) = u + v ,
(u) = u .
And for fuzzy sets, we adopt the most common metric dr (d∞ ) (see [12,20,22]), viewed as a generalization of the
Hausdorff metric from C(S) to F(S), where
dr (u, v) =
1
0
1/r
r
dH
(u , v ) d
d∞ (u, v) = sup d H (u , v ),
∈(0,1]
if 1r < ∞,
(2.3)
(2.4)
where u, v ∈ F(S). And the norm of u ∈ F(S) is defined as ur = dr (u, 0) or u∞ = d∞ (u, 0) = u 0 r .
The concept of a fuzzy random set as a generation for a random set was extensively studied by Puri and Ralescu
[22]. A fuzzy random set is a function X : → F(S) such that for each ∈ (0, 1], X () = {x ∈ S; X ()(x) } is
a random set in S (cf. [18]). A random fuzzy set X is said to be integrably bounded if the real-valued random variable
K.-A. Fu, L.-X. Zhang / Fuzzy Sets and Systems 159 (2008) 3360 – 3368
3363
supp u is integrable. The expectation of a fuzzy random set X, denoted by E[X ], is an element in F(S) such that for
each ∈ (0, 1],
X dP = cl{E( f ); f ∈ S X },
(E[X ]) = cl
where the closure is taken in S and S X = { f ∈ L 1 (, S); f () ∈ X () a.e.}. By virtue of the existence theorem (cf.
[18]), we have an equivalent definition as follows:
E[X ](x) = sup{ ∈ (0, 1]; x ∈ E[X ]}.
Furthermore, (E[co X ]) = E[(co X ) ] for any ∈ (0, 1]. And fuzzy random sets {X n ; n 1} is said to be independent
if for any ∈ (0, 1], the sequence of random sets {X n ; n 1} is independent. For arrays of random sets or fuzzy
random sets, the rowwise independent can be defined for every row similarly.
Note that if {X ni ; 1i n, n 1} is an array of random vector in d , then {X ni ; 1i n, n 1} is said to be
uniformly integrable if for every > 0 there exists a ∈ such that supn,i EX ni I (X ni > a) < , where I M is an
indicator function of a set M. The set {x : xa} is compact in d . But for a space of infinite dimension (such as
C(S) and K (S)), a bounded closed set may not necessarily be compact. Hence, to obtain SLLNs for random compact
sets and fuzzy random sets in Banach space, we firstly introduce the following definitions.
Definition 2.1. A collection {X ni ; 1i n, n 1} of random compact sets in C(S) is said to be compact uniformly
integrable (CUI) if for every > 0 there exists a compact subset K of C(S) such that supn,i EX ni I (X ni ∈
/ K ) < .
Definition 2.2. A collection {X ni ; 1i n, n 1} of fuzzy random sets in F(S) is said to be CUI if for every > 0
/ K ) < .
there exists a compact subset K of C(S) such that for every ∈ (0, 1], supn,i EX ni I (X ni ∈
Remark 2.3. By the definitions above, if the fuzzy random sets {X ni ; 1i n, n 1} is CUI, then so is {X ni ;
1i n, n 1} for every ∈ (0, 1]. And for the role of the concept of compactly uniformly integrable in classical
probability, readers may refer to Billingsley [3] for details.
In order to prove the main results in the last section, here we introduce some lemmas which will be used later.
Rådström [23] showed that the collection of compact convex subsets of a Banach space can be embedded as a convex
cone in a normed linear space. That is, the metric space Cc (S) can be embedded in a separable Banach space N with
an isometry g : Cc (S) → N . Thus we have the following lemma.
Lemma 2.1 (Rådström [23]). Let X : → Cc (S) be a random compact set such that EX < ∞. If g : Cc (S) → N
is the isometry given by the Rådström embedding theorem, then we have E(g ◦ X ) = g(EX ).
Next lemma due to Taylor et al. [27] is for uniform convergence over compact subsets {Ani } of S which are in a d H
compact subset K.
Lemma 2.2 (Taylor et al. [27]). Let K be a compact subset of C(S) and {ani } be an array of nonnegative constants
such that
n
ani 1 and
i=1
Then
dH
n
i=1
max ani → 0 as n → ∞.
1i n
ani bni Ani ,
n
ani bni co Ani
→ 0 as n → ∞
i=1
for any array of {bni } consisting of 0’s and 1’s and for any array {Ani } ⊂ K .
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K.-A. Fu, L.-X. Zhang / Fuzzy Sets and Systems 159 (2008) 3360 – 3368
3. Strong laws of large numbers
In this section, we discuss the SLLNs, and in the sequel let {X ni } be an array of integrably bounded random compact
sets or fuzzy random sets. It is known that Hu et al. [8] obtained the SLLNs for arrays of independent and identically
2 < ∞. They also extended their results to
distributed random variables {X ni } with mean zero by assuming that EX 11
arrays of rowwise independent, but not necessarily identically distributed random variables with mean zero by requiring
that they were stochastically dominated by a random variables X in the sense that there exists a random variable X such
that P(|X ni | t)P(|X |t) for all n, i and all t > 0 and EX 2 < ∞. Similar results will now be obtained for arrays
of rowwise independent (not necessary identically distributed) random compact sets and fuzzy random sets.
Theorem 3.1. Let {X ni ; 1i n, n 1} be an array of rowwise independent and CUI random compact sets in C(S).
Suppose there exists a random variable X such that P(X ni t)P(|X |t) for all n, i and all t > 0 and EX 2 < ∞.
Then for every > 0,
n
∞
n
1
1
(3.1)
P dH
X ni ,
E co X ni 7 < ∞.
n
n
n=1
i=1
i=1
In particular,
n
n
1
1
dH
X ni ,
E co X ni → 0 a.s.
n
n
i=1
i=1
Proof. For every > 0, we can choose a compact subset K of C(S) such that for all n, i
EX ni I (X ni ∈
/ K ) .
Since K is compact, there exist k1 , . . . , km ∈ K such that
K ⊆
m
{z : z − k j < } ≡:
j=1
m
B(k j , ).
j=1
I (X ∈ K ), where
Set Z ni = Z ni
ni
Z ni
:= k1 I [X ni ∈ B(k1 , )] +
m
j=2
⎧
⎨
kj I
⎩
X ni ∈ B(k j , ) ∩
j−1
B(kl , )c
l=1
⎫
⎬
⎭
.
Thus Z ni is a random compact set taking finitely many values, and the sequence {Z ni ; 1i n, n 1} is also rowwise
independent. Hence we have that
n
n
n
n
1
1
1
1
dH
X ni ,
E co X ni d H
X ni ,
X ni I (X ni ∈ K )
n
n
n
n
i=1
i=1
i=1
i=1
n
n
1
1
+d H
X ni I (X ni ∈ K ),
Z ni
n
n
i=1
⎛ i=1
⎞
m
m
n n 1
1
+d H ⎝
k j I (Z ni = k j ),
k j P(Z ni = k j )⎠
n
n
i=1 j=1
i=1 j=1
⎛
⎞
n
n m
m
1
1
k j P(Z ni = k j ),
co k j P(Z ni = k j )⎠
+d H ⎝
n
n
i=1 j=1
i=1 j=1
n
n
1
1
+d H
E co Z ni ,
E co X ni I (X ni ∈ K )
n
n
i=1
i=1
K.-A. Fu, L.-X. Zhang / Fuzzy Sets and Systems 159 (2008) 3360 – 3368
+d H
3365
n
n
1
1
E co X ni I (X ni ∈ K ),
E co X ni
n
n
i=1
i=1
=: (I1 ) + (I2 ) + (I3 ) + (I4 ) + (I5 ) + (I6 ),
m
I (X
where (I3 ) and (I4 ) follow from the notations Z ni = Z ni
ni ∈ K ) =
j=1 k j I (Z ni = k j ) and EI (Z ni = k j ) =
P(Z ni = k j ). Now we begin to deal with (I1 ).(I6 ), respectively. For (I1 ), we have
n
n
1
1
X ni ,
X ni I (X ni ∈ K )
dH
n
n
i=1
i=1
n
n
1
1
(X ni I (X ni ∈
/ K ) − EX ni I (X ni ∈
/ K )) +
EX ni I (X ni ∈
/ K )
n
n
i=1
i=1
n
1
(X ni I (X ni ∈
/ K ) − EX ni I (X ni ∈
/ K )) + n
i=1
and {X ni I (X ni ∈
/ K ) − EX ni I (X ni ∈
/ K ); 1i n, n 1} is an array of rowwise independent random variables
with means zero. Also we have that for all n, i,
/ K ) − EX ni I (X ni ∈
/ K ) t)P(X ni I (X ni ∈
/ K ) + t)P(|X | + t).
P(X ni I (X ni ∈
Thus from Theorem 2 of Hu et al. [8], it follows that
∞
P((I1 ) 2) < ∞.
n=1
For (I2 ), by the construction of Z ni , it follows that
n
n
n
1
1
1
X ni I (X ni ∈ K ),
Z ni d H (X ni I (X ni ∈ K ), Z ni ) < .
dH
n
n
n
i=1
i=1
i=1
For (I3 ), we have
(I3 )
m
j=1
n
1 k j [I (Z ni = k j ) − P(Z ni = k j )] .
n
i=1
Notice that {I {Z ni = k j } − P(Z ni = k j )} are bounded random variables. Let = /(
Hu et al. [8] again, we have
n
∞
∞
m 1 P((I3 ) )
P [I (Z ni = k j ) − P(Z ni = k j )] < ∞.
n
n=1
j=1 n=1
i=1
For (I4 ), from Lemma 2.2 it follows that for n large enough
n
m
n
1
1
dH
k j P(Z ni = k j ),
co k j P(Z ni = k j ) < .
(I4 )
n
n
j=1
i=1
i=1
For (I5 ), by the construction of Z ni , it leads to
n
1
d H (E co Z ni , E co X ni I (X ni ∈ K ))
(I5 ) n
i=1
n
1
Ed H (Z ni , X ni I (X ni ∈ K )) < .
n
i=1
m
j=1 k j ).
By Theorem 2 of
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K.-A. Fu, L.-X. Zhang / Fuzzy Sets and Systems 159 (2008) 3360 – 3368
For (I6 ), we have
(I6 )
n
n
1
1
d H (E co X ni I (X ni ∈
/ K ), 0)
EX ni I (X ni ∈
/ K ).
n
n
i=1
i=1
Now by combining the arguments above, it yields
n
∞
n
1
1
P dH
X ni ,
E co X ni 7 < ∞
n
n
n=1
i=1
i=1
and the proof is now completed. Since tightness and a uniformly bounded pth ( p > 1) moment condition imply compact uniform integrability, the
following corollary can be obtained easily.
Corollary 3.2. Let {X ni ; 1i n, n 1} be a tight array of rowwise independent random compact sets in C(S)
satisfying supn,i EX ni 2+ < ∞ for some > 0. Then
n
n
1
1
X ni ,
E co X ni → 0 a.s.
dH
n
n
i=1
i=1
Now we begin to state SLLNs for fuzzy random sets. Notice that Krätschmer [14] also investigated the laws of large
numbers for fuzzy random sets in Euclidean spaces. His proofs heavily relied on an appropriate identifications of fuzzy
random sets with random elements in Banach spaces satisfying convexity property, that is, in order to apply some
known convergence results, he took fuzzy random sets as separable Hilbert-space-valued random variables. Here we
consider the SLLNs for arrays of fuzzy random sets in separable Banach spaces, and we also take a different approach
to deal with convergence results by imposing a uniformly boundness condition.
Theorem 3.3. Let {X ni ; 1i n, n 1} be an array of rowwise independent fuzzy random sets in F(S) which are CUI
and satisfy that for each > 0, there exists a partition 0 = 0 < 1 < · · · < m = 1 of [0, 1] such that
max Ed H (X ni +
k−1 , X ni k ) < for all n, i.
(3.2)
1k m
Suppose there exists a random variables X with EX 2 < ∞ such that for all n, i,
P(X ni ∞ t)P(|X |t) for all t > 0.
Then
d∞
n
n
1
1
X ni ,
EX ni
n
n
i=1
(3.3)
→ 0 a.s.,
i=1
where d∞ is defined in (2.4).
Proof. First, we choose a partition 0 = 0 < 1 < · · · < m = 1 of [0, 1] such that for all n, i,
max Ed H (X ni +
k−1 , X ni k ) < /2.
(3.4)
1k m
Thus it follows that
n
n
1
1
sup
dH
X ni ,
E co X ni n
n
k−1 < k
i=1
i=1
n
n
n
n
1
1
1
1
dH
sup
X ni ,
X ni k + d H
X ni k ,
E co X ni k
n
n
n
n
k−1 < k
i=1
i=1
i=1
i=1
K.-A. Fu, L.-X. Zhang / Fuzzy Sets and Systems 159 (2008) 3360 – 3368
3367
n
n
1
1
+ dH
E co X ni k ,
E co X ni n
n
i=1
i=1
n
n
n
1
1
1
+
d H (X ni k−1 , X ni k ) + d H
X ni k ,
E co X ni k
n
n
n
i=1
i=1
i=1
n
1
+
d H (E co X ni k , E co X ni +
k−1 )
n
i=1
=: (I I1 ) + (I I2 ) + (I I3 ),
where the second inequality follows from (2.2).
+
For (I I1 ), notice that {d H (X ni +
k−1 , X ni k ) − Ed H (X ni k−1 , X ni k ); 1i n, n 1} is an array of rowwise independent random variables with means zero and there exists a random variable X satisfying EX 2 < ∞ and
+
P(|d H (X ni +
k−1 , X ni k ) − Ed H (X ni k−1 , X ni k )|t)P(2X ni ∞ + t)P(2|X | + t).
Thus it follows from Theorem 2 in Hu et al. [8] and (3.4) that for n large enough
n
n
1
1
+
+
(I I1 ) =
[d H (X ni k−1 , X ni k ) − Ed H (X ni k−1 , X ni k )] +
Ed H (X ni +
k−1 , X ni k ) < a.s.
n
n
i=1
i=1
For (I I2 ), since {X ni ; 1i n, n 1} is an array of CUI fuzzy random sets, it follows from Remark 2.3 that
{X ni ; 1i n, n 1} also is CUI for each ∈ (0, 1]. Thus coupled with
P(X ni t)P(X ni ∞ t)P(|X |t),
it follows from Theorem 3.1 that (I I2 ) → 0 a.s. as n → ∞.
For (I I3 ), it is readily seen that
(I I3 )
n
1
Ed H (X ni k , X ni +
k−1 ) < /2.
n
i=1
Hence, for n large enough, it leads to
n
n
n
n
1
1
1
1
d∞
X ni ,
E co X ni = max
sup
dH
X ni ,
E co X ni 1 k m k−1 < k
n
n
n
n
i=1
i=1
< 3/2.
i=1
i=1
Thus we get the desired result by the arbitrariness of . For fuzzy random sets in F(S), when the stochastic domination condition is replaced with a more restrictive
condition, we have the following similar result.
Theorem 3.4. Let {X ni ; 1i n, n 1} be an array of rowwise independent fuzzy random sets in F(S) which are
uniformly bounded and CUI. Then
n
n
1
1
X ni ,
E co X ni → 0 a.s.,
dr
n
n
i=1
i=1
where r is defined as in (2.3).
Proof. Notice that for each ∈ (0, 1], {X ni } is CUI by hypothesis, and the uniform boundness of {X ni } implies
X ni M. Thus it follows from Theorem 3.1 that for each ∈ (0, 1],
n
n
1
1
r
dH
X ni ,
E co X ni → 0 a.s.,
n
n
i=1
i=1
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K.-A. Fu, L.-X. Zhang / Fuzzy Sets and Systems 159 (2008) 3360 – 3368
which, coupled with the bounded convergence theorem, leads to
n
n
1
1
X ni ,
E co X ni → 0 a.s.
dr
n
n
i=1
i=1
Remark 3.1. When S = d , it is well-known that closure and boundness was sufficient for CUI in C(d ). Hence, for
applications in F(d ), the condition of CUI can be replaced by moment conditions.
Acknowledgements
The authors wish to express their deep gratitude to the editor and the anonymous referees for their valuable comments
on an earlier version which improve the quality of this paper, and thanks also to Dr. V. Krätschmer for sending us the
unpublished manuscript.
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