# Ideal Convergence of Random Variables

Authors: B. Hazarika; S. A. Mohiuddine
Journal: Journal of Function Spaces and Applications
 (2013)

---

## Abstract

The aim of this paper is to introduce and study the notion ofI-convergence of random variables via probabilistic norms. Furthermore, we introduce I-convergence in Lp space and establish some interesting results.


---

## Body

## 1. Introduction

Fast [1] and Steinhaus [2] independently introduced the notion of statistical convergence for sequences of real numbers, which is a generalization of the concept of convergence. The concept of statistical convergence is a very useful functional tool for studying the convergence problems of numerical sequences through the concept of density. Later on, several generalizations and applications of this concept have been presented by various authors (see [3–10] and references therein). Kostyrko et al. [11] presented a generalization of the concept of statistical convergence with the help of ideal I of subsets of the set of natural numbers ℕ and further studied in [12–16].Menger [17] presented an interesting and important generalization of the concept of a metric space under the name of statistical metric space by using probability distribution function, which is now called a probabilistic metric space. By using the concept of Menger, Šerstnev [18] introduced the concept of probabilistic normed space (for random normed space, see [19]), which is an important generalization of deterministic results of linear normed spaces. Afterward, Alsina et al. [20] presented a new definition of probabilistic normed space which includes the definition of Šerstnev as a special case.The concept of ideal convergence for single and double sequence of real numbers in probabilistic normed space was introduced and studied by Mursaleen and Mohiuddine [21, 22]. In the recent past, Mursaleen and Alotaibi [23] and Mohiuddine et al. [24] studied the notion of ideal convergence for single and double sequences in random 2-normed spaces, respectively. For more detail and related concept, we refer to [25–33] and references therein.


## 2. Basic Definitions and Notations

The notion of statistical convergence depends on the density (asymptotic or natural) of subsets ofℕ. A subset E of ℕ is said to have natural density δ(E) if
(1)δ(E)=limn→∞1n|{k≤n:k∈E}|exists.A sequencex=(xk) is said to be statistically convergent [1] to ℓ if for every ε>0(2)δ({k∈ℕ:|xk-ℓ|≥ε})=0.
In this case, we write S-limx=ℓ or xk→ℓ(S), and S denotes the set of all statistically convergent sequences.An ideal is defined as a hereditary and additive family of subsets of a nonempty arbitrary setX; here, in our study, it suffices to take I as a family of subsets of ℕ, positive integers; that is, I⊂2ℕ, such that ϕ∈I, A∪B∈I for each A,B∈I, and each subset of an element of I is an element of I. A nonempty family of sets ℱ⊂2ℕ is a filter on ℕ if and only if ϕ∉ℱ, A∩B∈ℱ for each A,B∈ℱ, and any superset of an element of ℱ is in ℱ. An ideal I is called nontrivial if I≠ϕ and ℕ∉I. Clearly, I is a nontrivial ideal if and only if ℱ=ℱ(I)={ℕ-A:A∈I} is a filter in ℕ, called the filter associated with the ideal I. A nontrivial ideal I is called admissible if and only if {{n}:n∈ℕ}⊂I. A nontrivial ideal I is maximal if there cannot exist any nontrivial ideal J≠I containing I as a subset. Further details on ideals can be found in Kostyrko et al. (see [11]). Recall that a sequence x=(xk) of points in ℝ is said to be I-convergent to a real number ℓ if {k∈ℕ:|xk-ℓ|≥ε}∈I for every ε>0 (see [11]). In this case, we write I-limxk=ℓ.Now, we recall some notations and basic definitions that we are going to use in this paper.We use the notion and terminology of [34]. Thus, Δ+ is the space of probability distribution functions F that are left continuous on ℝ+=(0,+∞), F(0)=0, and F(+∞)=1. The space Δ+ is partially ordered by the usual pointwise ordering of functions and has both a maximal element ε0 and a minimal element ε∞; these are given, respectively, by
(3)ϵ0(t)={0,ift≤01,ift>0,ϵ∞(t)={0,ift<+∞1,ift=+∞.
There is a natural topology on Δ+ that is induced by the modified Lévy metric dL (see, [34, 35]); that is,
(4)dL(F,G)=inf{h:both[F,G;h]and[G,F;h]hold}
for all F,G∈Δ+ and h∈(0,1], where [F,G;h] denote the condition
(5)G(t)≤F(t+h)+h,fort∈(0,1h).
Convergence with respect to this metric is equivalent to weak convergence of distribution functions, that is (Fn) in Δ+ converges weakly to F in Δ+ (written as Fn→wF) if and only if (Fn(t)) converges to F(t) at every point of continuity of the limit function F. Consequently, we have
(6)Fn⟶wFiffdL(Fn,F)⟶0,(7)F(x)>1-xiffdL(F,ε0)<xforeveryx>0.Moreover, the metric space(Δ+,dL) is compact.Definition 1.
Atriangular norm (or briefly, t-norm) is a binary operation T:[0,1]×[0,1]→[0,1] that satisfies the following conditions (see [36]):(TN1)
T(s,t)=T(t,s) for all s,t∈[0,1] (T is commutative),(TN2)
T(T(s,t),u)=T(s,T(t,u)) for all s,t,u∈[0,1] (T is associative),(TN3)
T(s,t)≤T(s′,t) for all t∈[0,1] whenever s≤s′ (T is nondecreasing),(TN4)
T(1,t)=t for every t∈[0,1] (T satisfies the boundary condition).T* is a continuous t-conorm, namely, a continuous binary operation on [0,1] that is related to a continuous t-norm through T*(s,t)=1-T(1-s,1-t).Notice that by virtue of its commutativity, anyt-norm T is nondecreasing in each place. Some examples of t-norms T and its t-conorms T* are W(x,y)=max{x+y-1,0}, M(x,y)=min{x,y}, and Π(x,y)=x·y and W*(x,y)=min{x+y,1}, M*(x,y)=max{x,y}, and Π*(x,y)=x+y-x·y.Definition 2.
A triangle function is a binary operation onΔ+, namely, a function τ:Δ+×Δ+→Δ+ that is associative, commutative, and nondecreasing and which has ε0 as unit; that is, for all F,G,H,K∈Δ+, one has:(1)
τ(F,ϵ0)=F;(2)
τ(F,G)=τ(G,F);(3)
τ(F,G)≤τ(H,K) whenever F≤H,G≤K;(4)
τ(τ(F,G),H)=τ(F,τ(G,H)).Particular and relevant triangle functions are the functionsτT,τT* and those of the form ΠT which, for any continuous t-norm T and any x>0, are given by
(8)τT(F,G)(x)=sup{T(F(u),G(v)):u+v=x},τT*(F,G)(x)=inf{T*(F(u),G(v)):u+v=x},ΠT(F,G)(x)=T(F(x),G(x)).Definition 3.
A probabilistic normed space (or briefly, PN space) is a quadruple(X,ν,τ,τ*), where X is a real linear space, τ and τ* are continuous triangle functions such that τ≤τ*, and the mapping ν:X→Δ+ called the probabilistic norm, for all p and q in X, satisfies the following conditions:  (PN1)
νp=ϵ0 if and only if p=θ (θ is the null vector in X);(PN2)
for allp∈X,ν-p=νp;(PN3)
νp+q≥τ(νp,νq);(PN4)
for alla∈[0,1], νp≤τ*(νap,ν(1-a)p).If a PN space(X,ν,τ,τ*) satisfies the following condition:  (Š) for allp∈X, for all λ∈ℝ∖{0}, for all t>0, νλp(t)=νp(t/|λ|),then it is called a Šerstnev space; the condition (Š) implies that the best-possible selection for τ* is τ*=τM, which satisfies a stricter version of (PN4); namely,
(9)∀a∈[0,1],νp=τM(νap,ν(1-a)p).
A Šerstnev space is denoted by (X,ν,τ), since the role of τ* is placed by a fixed triangle function τM, which satisfies (PN2).A PN spaceX is endowed with the strong topology (briefly 𝒮-topology) generated by the strong neighborhood system {𝒩θ(λ):λ>0}, where
(10)𝒩θ(λ)={p∈X:dL(vp,ε0)<λ}
determines a first countable and Hausdorff topology on X (see [34]), and it is metrizable.The following lemma is an immediate consequence of the definition of neighborhood of zero and (7).Lemma 4.
In a PN space(X,ν,τ,τ*), for each p∈X, one has
(11)νp(λ)>1-λ⟺p∈𝒩θ(λ).A sequence(pn) of elements in X converges to θ, the null element of X, in the strong topology (briefly 𝒮-topology) (written pn→θ) if and only if
(12)limn→∞dL(νpn,ε0)=0.
That is, for every λ>0, there is an integer m=m(λ)∈ℕ such that dL(νpn,θ)<λ for all n≥m, where dL is  defined in (4). In terms of neighborhood, we have pn→θ provided that for any λ>0 there is an N(λ)∈ℕ such that pn∈𝒩θ(λ)  (i.e.,  νpn(λ)>1-λ) whenever n≥N. In this case, we write pk→𝒮θ or 𝒮-limkpk=θ. Thus, the 𝒮-topology can be completely specified by means of 𝒮-convergence of sequences.A sequence(pk) is said to be 𝒮-Cauchy if for any λ>0, there exists an integer N(λ)∈ℕ such that pk-pn∈𝒩θ(λ) whenever k,n≥N(λ).Lemma 5 (see [37]).
For anyα∈ℝ, any p∈X, and any ε>0, there exists a λ>0 such that
(13)αp∈𝒩θ(ε)wheneverp∈𝒩θ(λ).Lemma 6 (see [37]).
If0≤α<1, then for any p∈X(14)ναp≥νp.Lemma 7 (see [37]).
For anyα∈ℕ, any p∈X, and any t>0, there is a λ>0 such that
(15)dL(ναp,ε0)<twheneverdL(νp,ε0)<λ.We observe that, in view of Lemma4 and (PN3), we have the following lemma.Lemma 8.
Let(X,ν,τ,τ*) be a PN space. For all p,q,r∈X,
(16)dL(νp-r,ε0)≤dL(τ(νp-q,νq-r),ε0).An important class of PN spaces is that ofE-normed spaces (see [38]). Let (Ω,𝒜,P) be a probability space, (X,∥·∥) a normed space, and S a linear space of X-valued random variables (possibly, the entire space). For every p∈S and for every λ∈ℝ, let ν:S→Δ+ be defined by
(17)νλ:=P({ω∈Ω:∥p(ω)∥<λ});
then (S,ν) is an E-normed space (briefly, EN space) with the base (Ω,𝒜,P) and target (X,∥·∥).Example 9.
LetL0=L0(Ω,𝒜,P), the linear space of (equivalence classes of) random variable f:Ω→ℝ. Let ν:S→Δ+ be defined, for every f∈L0 and for every λ∈ℝ, by
(18)νf(λ)=P({ω∈Ω:|f(ω)|<λ}).
Then, the couple(L0,ν) is an EN space. It is a PN space under the triangle function τW and τM (see [34]).


## 3. Ideal Convergence of Random Variables

Throughout the paper, we denoteI as an admissible ideal of subsets of ℕ, unless otherwise stated. In this section, we begin with the definition of ideal convergence of probability distribution functions.Definition 10.
LetI⊂2ℕ, and let (Δ+,dL) be a Lévy metric space. A sequence (Fk) in Δ+ is said to be I-convergent (weakly) to F∈Δ+ if and only if for every λ>0, the set
(19){k∈ℕ:dL(Fk,F)≥λ}∈I,
or
(20){k∈ℕ:dL(Fk,F)<λ}∈ℱ.
In this case, we write Fk→𝒲IF or 𝒲I-limFk=F.By (7) and (19), the following lemma can be easily verified.Lemma 11.
Let(Δ+,dL) be a Lévy metric space and (Fk) a sequence in Δ+. Then, for every λ>0, the following statements are equivalent:  (i)
Fk→𝒲Iε0,(ii)
{k∈ℕ:dL(Fk,ε0)<λ}∈ℱ,(iii)
{k∈ℕ:Fk(λ)>1-λ}∈ℱ,(iv)
𝒲I-limkdL(Fk,ε0)=0.Definition 12.
Let(X,ν,τ,τ*) be a PN space. A sequence (pk) in X is said to be I-convergent toθin the strong topology (or strong-I-convergent) if and only if for every λ>0, the set
(21){k∈ℕ:dL(νpk,ε0)≥λ}∈I,
or
(22){k∈ℕ:dL(νpk,ε0)<λ}∈ℱ.
In this case, we write pk→𝒮Iθ or 𝒮I-limpk=θ, where θ is called the 𝒮I-limit of (pk). In terms of neighborhoods, we have
(23)pk⟶𝒮Iθiff{k∈ℕ:pk∉𝒩θ(λ)}∈I.
The following lemma is an immediate consequence of the above definition.Lemma 13.
Let(X,ν,τ,τ*) be a PN space and (pk) a sequence in X. Then, for every λ>0, the following statements are equivalent:  (i)
pk→𝒮Iθ,(ii)
{k∈ℕ:pk∉𝒩θ(λ)}∈I,(iii)
{k∈ℕ:dL(νpk,ε0)≥λ}∈I.Theorem 14.
Let(X,ν,τ,τ*) be a PN space, and if a sequence (pk) in X is 𝒮I-convergent, then 𝒮I-limpk is unique.Proof.
Suppose that𝒮I-limpk=p and 𝒮I-limpk=q with p≠q. Then, for λ>0, define the following sets:
(24)K1(λ)={k∈ℕ:pk-p∉𝒩θ(λ)},K2(λ)={k∈ℕ:pk-q∉𝒩θ(λ)}.
Since𝒮I-limpk=p, using Lemma 13, we have K1(λ)∈I. Also, using 𝒮I-limpk=q, we get K2(λ)∈I. Let
(25)K(λ)=K1(λ)∪K2(λ).
Then, K(λ)∈I for all λ>0. This implies that its complement Kc(λ) is a nonempty set in ℱ for all λ>0. Now, if k∈Kc(λ), then pk-p∈𝒩θ(λ). Let dL(νpk-p,ε0)=α. Then, λ-α>0, and the uniform continuity of τ implies that there exists a λ1>0 such that
(26)dL(τ(νpk-p,G),νpk-p)<λ-α
whenever dL(G,ε0)<λ1. Now, let pk-q∈𝒩θ(λ1), and then dL(νpk-q,ε0)<λ1. Thus, by (16), we have
(27)dL(νp-q,ε0)≤dL(τ(νpk-p,νpk-q),ε0)≤dL(τ(νpk-p,νpn-q),νpk-p)+dL(νpk-p,ε0)<λ-α+α=λ.
Hence,p-q∈𝒩θ(λ). Since λ is arbitrary, we get νp-q=ε0, which yields p-q=0; that is, p=q. Thus, this   completes the proof.The next theorem gives the algebraic characterization of𝒮I-convergence in PN space.Theorem 15.
Let(X,ν,τ,τ*) be a PN space and (pk) and (qk) two sequences in X.(a)
If𝒮I-limpk=p and c∈ℝ, then 𝒮I-limcpk=cp.(b)
If𝒮I-limpk=p and 𝒮I-limqk=q, then 𝒮I-lim(pk±qk)=p±q.Proof.
(a) Letc∈ℝ, and let (pn) be a sequence in X such that 𝒮I-limpk=p. Then, from (15), we have
(28){k∈ℕ:dL(νpk-p,ε0)<λ}⊂{k∈ℕ:dL(νc(pk-p),ε0)<t}
for any t>0. Since 𝒮I-limpk=p, we have
(29){k∈ℕ:dL(νpk-p,ε0)<λ}∈ℱ.
Thus, we have for each t>0(30){k∈ℕ:dL(νc(pk-p),ε0)<t}∈ℱ.
This shows that 𝒮I-limcpk=cp.
(b) Let(pk) and (qk) be two sequences in X such that 𝒮I-limpk=p and 𝒮I-limqk=q. Then, for λ>0, define the following sets:
(31)A(λ)={k∈ℕ:pk-p∉𝒩θ(λ)},B(λ)={k∈ℕ:qk-q∉𝒩θ(λ)}.
Now, we can write
(32)ν(pk+qk)-(p+q)≥τ(νpk-p,νqk-q),
and hence,
(33)dL(ν(pk+qk)-(p+q),ε0)≤dL(τ(νpk-p,νqk-q),ε0)pppppppforeveryk∈ℕ.
By uniform continuity of τ, we can say that for any t>0 there exists a λ>0 such that dL(τ(F,G),ε0)<t whenever dL(F,ε0)<λ and dL(G,ε0)<λ, where F,G∈Δ+. Now, let t>0. Then, we can find a λ>0 such that
(34)dL(τ(νpk-p,νqk-q),ε0)<t;
that is,
(35)dL(ν(pk+qk)-(p+q),ε0)<t
whenever pk-p∈𝒩θ(λ) and qk-q∈𝒩θ(λ), that is, dL(νpk-p,ε0)<λ and dL(νqk-q,ε0)<λ. Thus, we have
(36){k∈ℕ:dL(ν(pk+qk)-(p+q),ε0)≥t}⊆{k∈ℕ:pk-p∉𝒩θ(λ)}∪{k∈ℕ:qk-q∉𝒩θ(λ)}
for each t>0.
Then, for eacht>0, we have
(37){k∈ℕ:dL(ν(pk+qk)-(p+q),ε0)≥t}⊆A(λ)∪B(λ).
Since I is admissible, from (37), we have
(38){k∈ℕ:dL(ν(pk+qk)-(p+q),ε0)≥t}∈Iforeacht>0.
Hence, 𝒮I-lim(pk+qk)=p+q.
Similarly, we can show that𝒮I-lim(pk-qk)=p-q.Theorem 16.
Let(X,ν,τ,τ*) be a PN space, and let (pk) be a sequence in X. If 𝒮-limpk=p, then 𝒮I-limpk=p.Proof.
Let𝒮-limpk=p; then for every λ>0 there exists an integer N=N(λ)∈ℕ such that
(39)pk-p∈𝒩θ(λ)wheneverk≥N(λ).
Therefore, the set(40)B(λ)={k∈ℕ:pk-p∉𝒩θ(λ)}⊆{1,2,…,N-1}.
But, with I being admissible, we have B(λ)∈I. Hence, 𝒮I-limpk=p.Theorem 17.
Sequential method𝒮I is regular.Proof.
The proof follows from the fact thatI is admissible and from Theorem 16.Theorem 18.
Let(X,ν,τ,τ*) be a PN space. A sequence (pk) in X is 𝒮I-convergent to θ if and only if there exists a subset K={i}⊂ℕ such that K∈ℱ and 𝒮-limk∈K,k→∞pk=θ.Proof.
Suppose that𝒮I-limpk=θ. Then, for j=1,2,3,…, we define the following set:
(41)A(j)={k∈ℕ:pk∈𝒩θ(1j),j∈ℕ}.
Since𝒮I-limpk=θ, it follows that Ac(j)∈I.
Now, forj=1,2,3,…, we observe that A(j)⊃A(j+1) and
(42)A(j)∈ℱ.
We show that, fork∈A(j), 𝒮-limk→∞pk=θ. Suppose that, for k∈A(j),(pk) is not 𝒮-convergent to θ. Then, there exists some i>0 such that
(43){k∈ℕ:pk∉𝒩θ(i)}
for infinitely many terms pk. Let
(44)A(i)={k∈ℕ:pk∉𝒩θ(i)}
and i>1/j, j=1,2,3,…. Then, we have A(i)∈I. Also, A(j)⊂A(i) implies that A(j)∈I, which contradicts (42) as A(j)∈ℱ. Hence, 𝒮-limk→∞pk=θ.Converse part is easy and can be omitted.


## 4. Ideal Convergence in Probability and inLp Space

Let(xk) be a sequence of random variables defined on a probability space (Ω,𝒜,P) taking values in a separable normed space (X,∥·∥), where ∥·∥ is the norm. Then, we say that a sequence (xk)  converges in probability or converges in measure to θ if for every λ>0,
(45)limkP(∥xk∥>λ)=0.Equivalently, for anyλ>0, there is an integer n0∈ℕ such that
(46)P(∥xk∥<λ)>1-λ,∀k≥n0.In this case, we writexk→𝒫θ.Now, we give the definition of ideal convergence in probability as follows.Definition 19.
A sequence(xk) of random variables is said to be I-convergent in probability to θ, if, for every λ>0, the set
(47){k∈ℕ:P(∥xk∥<λ)≤1-λ}∈I
or
(48){k∈ℕ:P(∥xk∥<λ)>1-λ}∈ℱ.
In this case, we writexk→I(𝒫)θ or I(𝒫)-limxk=θ.Example 20.
LetΩ=[0,1] and P a Lebesgue measure on [0,1]. Define the sequence of random variables xk=xk(t) for t∈[0,1] as follows:
(49)xk(t)={ℓ,ift∈[0,1k];0,otherwise.
For any ε>0, we have
(50)P(∥xk-0∥>ε)=P(xk(t)=ℓ)=P(t∈[0,1k])=lengthoftheinterval=1k⟶0.
It means that (xk) is I-convergent in probability to zero. That is, xk→I(𝒫)0.Theorem 21.
Let(fk) be a sequence of (equivalence classes of) E-valued random variables. Then, the following are equivalent:  (i)
fk→I(𝒫)θS;(ii)
νfk→𝒲Iε0;(iii)
fk→IθS  in theŠerstnev space(L0,ν,τW).Proof.
By definition, it is clear that (ii) and (iii) are equivalent, and it suffices to establish the equivalence of (i) and (ii).
LetK(λ)={k∈ℕ:P(∥fk∥<λ)≤1-λ}. We note that fk→I(𝒫)θS if and only if K(λ)∈I. But ℕ∖K(λ)∈ℱ. Therefore, for every k∈ℕ∖K(λ), we have P(∥fk∥<λ)>1-λ.  By (17), implies that νfk(λ)>1-λ.  By the property of 𝒮-topology, we have
(51){k∈ℕ:dL(νfk,ε0)<λ}⊇{k∈ℕ:νfk(λ)>1-λ}⊇{k∈ℕ:P(∥fk∥<λ)>1-λ};
that is,
(52){k∈ℕ:dL(νfk,ε0)≥λ}⊆{k∈ℕ:P(∥fk∥<λ)≤1-λ}.
Sincefk→I(𝒫)θS, therefore {k∈ℕ:P(∥fk∥<λ)≤1-λ}∈I.
Thus,{k∈ℕ:dL(νfk,ε0)≥λ}∈I; hence νfk→𝒲Iε0.In order to consider ideal convergence inLp with p∈[1,+∞), the following result connecting the Lp norms ∥·∥p with the probabilistic norm (17) will be needed (see [38]).Theorem 22.
LetLp=Lp(Ω,𝒜,P):={f∈L0:∫Ω‍|f|pdP<+∞} for p∈[1,+∞) and L∞:={f∈L0:∥f∥∞:=sup|f|<+∞}. If the probabilistic norm ν:L0→Δ+ is defined by
(53)νf(λ):=P({ω∈Ω:|f(ω)|<λ}),λ>0,
then for every f∈Lp,∥f∥p=(∫ℝ+‍λpdνf(λ))1/p, and for every f∈L∞,∥f∥∞=sup{t>0:νf(t)<1}.With the help of  Theorem22, one can characterize ideal convergence in Lp.Theorem 23.
Let(fk) be a sequence of (equivalence classes of) E-valued random variables in Lp. Then, the following statements are equivalent.
Ifp∈[1,+∞),(i)
fk→I(Lp)θS;(ii)
the sequence of the pth moments of the probabilistic norms(νfk)I-converges to 0.
Ifp=+∞,(iii)
fk→I(L∞)θS;(iv)
for everyλ>0,
{k∈ℕ:νfk(λ)=1}∈ℱ.Proof.
(i)⇔(ii) We note that fk→I(Lp)θS⇔{k∈ℕ:∥fk∥p≥λ}∈I for every λ>0. But
(54){k∈ℕ:(∫ℝ+‍tpdνfk(t))1/p≥λ}={k∈ℕ:∥fk∥p≥λ}.
Hence,(55){k∈ℕ:∫ℝ+‍tpdνfk(t)≥(λ)p}∈I.
(iii)⇒(iv) Suppose that fk→I(L∞)θS, that is, ∥fk∥∞→I(L∞)0, and let t>0, then, for every 0<ε<t, we have
(56){k∈ℕ:sup{t>0:νfk(t)<1}≥ε}={k∈ℕ:∥fk∥∞≥ε}∈I.
This implies that(57){k∈ℕ:sup{t>0:νfk(t)<1}<ε}∈ℱ.
Hence,(58){k∈ℕ:νfk(t)≥νfk(ε)=1}∈ℱ.
(iv)⇒ (iii) For t>0, suppose that {k∈ℕ:νfk(t)=1}∈ℱ, and therefore
(59){k∈ℕ:∥fk∥∞<t}∈ℱ,
which implies that ∥fk∥∞→I(L∞)0.



---

*Source: 148249-2013-10-30.xml*