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Generalized weighted random convergence in probability
Applied Mathematics and Computation 249 (2014) 502–509
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Generalized weighted random convergence in probability Sanjoy Ghosal Department of Mathematics, Kalyani Government Engineering College, Nadia, Kalyani 741235, West Bengal, India
a r t i c l e
i n f o
Keywords: Weighted statistical convergence Weighted k-statistical convergence Weighted modulus ab-statistical convergence of order c Weighted modulus ab-strong Cesáro convergence of order c Weighted modulus Sab -convergence of order c Weighted modulus N ab -convergence of order c
a b s t r a c t The definition of weighted statistical convergence was first introduced by Karakaya & Chishti (2009) and later on Mursaleen et al. (2012) modified the definition of this concept. Using the modified definition, Edely et al. (2013) had researched further in approximation theory for periodic functions and Belen & Mohiuddine (2013) had further extended to the new-definition of weighted k-statistical convergence. But some problems are still there, so the definition of weighted k-statistical convergence is needed to modify. In this paper, we will introduce some new constraints which will make the definition of weighted k-statistical convergence is more useful by using the definition of ab-statistical convergence. Using it and independently, some newly developed concepts of the convergence of a sequence of random variables in probability, namely, weighted modulus ab-statistical convergence of order c, weighted modulus ab-strong Cesáro convergence of order c, weighted modulus Sab -convergence of order c, and weighted modulus Nab -convergence of order c, have been introduced and their basic interrelations also have been investigated. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The concept of statistical convergence was introduced by Fast [10] and Steinhaus [32] and later on reintroduced by Schonberg [30] independently and is based on the notion of asymptotic density of the subset of natural numbers. However, the first idea of statistical convergence (by different name) was given by Zygmund [33] in the first edition of his monograph published in Warsaw in 1935. Later on it was further investigated from the sequence space point of view and linked with summability theorem by Fridy [12], Connor [6], Šalát [26], Gürdal [18], Das & Savas [7], Mursaleen [24], Fridy and Orhan [13]. In last five years, it has several devolvement and application of this convergence has been given by many author’s like: (i) statistical convergence of order a by Çolak [4] (statistical convergence of order a was also independently introduce by Bhunia et. al. [3]), (ii) k-statistical convergence of order a by Çolak and Bektasß [5], (iii) weighted statistical convergence by Mursaleen et. al. [25], (iv) weighted k-statistical convergence by Belen and Mohiuddine [2], (v) statistical convergence in probability by Ghosal [14,15], (vi) statistical convergence of a sequence of random function in probabilistic metric space by Sencimen [31], (vii) ab-statistical convergence by Aktug˘lu [1], (viii) A-statistical limit points via ideals by Gürdal and Sari [20], (ix) ideal convergence in n-norms space by Gürdal [19] and Gürdal and Huban [21], and so on. E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2014.10.056 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509
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The entire mathematical theory of probability was built by three objects, namely (a) the event space W, (b) the class of events M and (c) the probability function P : M!R. The order 3-tuples ðW; M; PÞ is called probability space. A mapping X of W to R is called a random variable or stochastic variable or a variate if for any x 2 R, the set
fx : 1 < XðxÞ 6 x; x 2 Wg 2 M: Ordinarily, the distribution of a random variable is required to obtain probability connected to the random variable. But Tchebycheff’s inequality indicates that, irrespective of the shape of the density curve
Pðm < X < m þ Þ > 1
r2 hold; 2
(where X has finite mean m and variance r), i.e., the probability that X takes values in the interval ðm ; m þ Þ centred at m is closed to 1, provided r is sufficiently small, i.e., we get and upper bound for the probability that the deviation of X from its mean is at least units in terms of variance of X and . No assumption on the distribution of X. From the practical point of view the discussion of a random variable X is highly significant if it is known that there exists a real constant x for which PðjX xj < Þ ’ 1, where > 0 is sufficiently small, that is, it is nearly certain that values of X lie in a very small neighbourhood of x. For a sequence of random variables fX n gn2N , each X n may not have the above property but it may happen that the aforesaid property (with respect to a real constant x) becomes more and more distinguishable as n gradually increases and the question of existence of such a real constant x can be answered by a concept of convergence in probability of the sequence fX n gn2N . In this paper the ideas of four types of convergences of a sequence of random variables, namely, (a) weighted modulus ab-statistical convergence of order c, (b) weighted modulus ab-strong Cesáro convergence of order c, (c) weighted modulus Sab -convergence of order c, (d) weighted modulus N ab -convergence of order c, have been introduced and the interrelations among them have been investigated. Also their certain basic properties have been studied. The main object of this paper is to modify the definition of weighted k-statistical convergence and to establish some important theorems related to the modes of convergences (a) to (d), which will effectively extend and improve all the existing results in this direction ([1–6,8–17,22–27]). Moreover, intend to establish the relations among these four summability notions. It is important to note that the methods of proofs and in particular the examples are not analogous to the real case. The following definitions and notions will be needed in sequel. Definition 1.1 (see [3,4]). A sequence fxn gn2N of real numbers is said to be statistically convergent of order c (where 0 < c 6 1) to x if for every > 0,
lim
1
n!1 nc
jfk 6 n : jxk xj P gj ¼ 0: Sc
In this case we write xn ! x and the set of all statistically convergent sequences of order c is denoted by Sc . Definition 1.2 (see [5]). Let fkn gn2N be a sequence of non-decreasing sequence of positive real numbers tending to 1 such that knþ1 6 kn þ 1; k1 ¼ 1. The sequence of real numbers fxn gn2N is said to be k-statistically convergent of order c (where 0 < c 6 1) to x if for any > 0,
lim
1 c
n!1 k n
jfk 2 In : jxk xj P gj ¼ 0; c
Sk
where In ¼ ½n kn þ 1; n. In this case we write xn ! x and the set of all k-statistically convergent sequences of order c is denoted by Sck . P Definition 1.3 (see [25]). Let ft n gn2N be a sequence of nonnegative real numbers such that t1 > 0 and T n ¼ nk¼1 t n !1 as n!1. Then the sequence of real numbers fxn gn2N is said to be weighted statistically convergent (or, SN -convergent) to x if for every > 0,
lim
1
n!1 T n
jfk 6 T n : tk jxk xj P gj ¼ 0: SN
In this case we write xn ! x and the set of all weighted statistically convergent sequences is denoted by SN .
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S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509
Definition 1.4 (see [2]). Let ft n gn2N be a sequence of nonnegative real numbers such that t1 > 0 and T kn ¼ k ; tÞ-summable to x if n!1. A sequence of real numbers fxn gn2N is said to be strongly ðN
P
k2In t k !1,
as
1 X t jx xj ¼ 0; k2In k k n!1 T k n lim
;t ½N k
where In ¼ ½n kn þ 1; n. In this case we write xn ! x and the set of all weighted k-statistically convergent sequences is k ; t. denoted by ½N Definition 1.5 (see [14]). Let fX n gn2N be a sequence of random variables, where each X n is defined on the same sample space W (for each n) with respect to a given class of events M and a given probability function P : M!R. The sequence fX n gn2N is said to be statistically convergent in probability to a random variable X (where X : W!R) if for any ; d > 0,
lim
1
n!1 n
jfk 6 n : PðjX k Xj P Þ P dgj ¼ 0: ðS;PÞ
In this case we write X n ! X and the class of all statistically convergent sequences of random variables in probability is denoted by ðS; PÞ. For more results on this convergence see the paper [8]. Definition 1.6 (see [1]). Let fan gn2N and fbn gn2N be two sequences of positive real numbers such that (i) a and b are both non-decreasing, (ii) bn P an ; 8n 2 N, (iii) bn an !1 as n!1. Then the sequence of real numbers fxn gn2N is said to be ab-statistically convergent of order c (where 0 < c 6 1) to x if for every > 0,
lim
n!1 ðb n
1 jfk 2 ½an ; bn : jxk xj P gj ¼ 0: an þ 1Þc c
Sab
In this case we write xn ! x and the set of all weighted statistically convergent sequences of order c is denoted by Scab . Definition 1.7 (see [23,27]). A modulus function / is a function from ½0; 1Þ to ½0; 1Þ such that (i) /ðxÞ ¼ 0 if and only if x ¼ 0, (ii) /ðx þ yÞ 6 /ðxÞ þ /ðyÞ, forall x; y > 0, (iii) / is increasing, (iv) / is continuous from the right at zero. A modulus function may be bounded or unbounded. Maddox [23], Savasß [27,28], Savasß and Patterson [29] and other authors used modulus function to construct new sequence spaces. 2. Main results Belen and Mohiuddine [2] first defined the concept of weighted k-statistical convergence as follows: Let ft n gn2N be a P sequence of nonnegative real numbers such that t1 > 0 and T kn ¼ k2In tk !1, as n!1 (where In ¼ ½n kn þ 1; n). Then the sequence of real numbers fxn gn2N is said to be weighted k-statistically convergent (or, SN k -convergent) to x if for every > 0,
lim
1
n!1 T k
jfk 6 T kn : t k jxk xj P gj ¼ 0:
n
SN
k
In this case we write xn ! x (or SN k lim xn ¼ x) and the set of all weighted k-statistically convergent sequences is denoted by SN k . Remark 2.4 (iv) (from [2]) shows that, if kn ¼ n for all n 2 N, then SN k -convergence coincides with SN -convergence. But the definition of weighted k-statistical convergence is not well defined in general. This follows from the following example. Example 2.1. Let tn ¼ n1p (where 0 < p 1), kn ¼ n, forall n 2 N and a sequence of real numbers fxn gn2N is defined by,
xn ¼
0; if n ¼ 2m; for any m 2 N; 1;
otherwise: SN
k
SN
It is quite clear that xn ! x (or, xn ! x) where x be any real number, i.e., for any bounded real sequence fxn gn2N is weighted k-statistically convergent (or weighted statistical convergence) to any real number. Hence the definition is not well defined. So the definition of weighted k-statistical convergence need to be modified. Now, we are going to modify the definition as follows:
S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509
505
P Definition 2.1. Let ftn gn2N be a sequence of real numbers such that lim inf n!1 tn > 0 and T abðnÞ ¼ k2½an ;bn tk , forall n 2 N. Then the sequence of real numbers fxn gn2N is said to be weighted ab-statistically convergent of order c (where 0 < c 6 1) to x if for every > 0,
1
lim
n!1 T
6 T abðnÞ : t k jxk xj P gj ¼ 0:
jfk c abðnÞ
c
ðSab ;t n Þ
In this case we write xn ! x. The class of all weighted ab-statistical convergence sequences of order c is denoted by ðScab ; tn Þ. Remark 2.1 (i) For c ¼ 1; an ¼ 1 and bn ¼ n, then weighted ab-statistical convergence of order c coincides with weighted statistical convergence. (ii) For c ¼ 1; an ¼ n kn þ 1 and bn ¼ n, then weighted ab-statistical convergence of order c coincides with weighted k-statistical convergence and so on. c
c
ðSab ;t n Þ
ðSab ;t n Þ
It is obvious that if xn ! x and xn ! y then x ¼ y. k ; tÞ-summable to x then it is weighted In Theorem 2.7 (i) (from [2]), say that ‘‘If a sequence fxn gn2N is strongly ðN k-statistical convergent to x.’’ Which is not true given by the following example. Example 2.2. Let tn ¼ n; kn ¼ n, where n 2 N and a sequence of real numbers fxn gn2N is defined by,
(
xn ¼
1 ; n
if n is an even integer; otherwise:
0;
Now let 0 < < 1. Then for the large value of n we get, Tn 1X 2 1 1 and t k jxk 0j 6 jfk 6 T n : t k jxk 0j P gj P 3 P : T n k2I nþ1 Tn 3 Tn n
k ;t ½N
So xn ! 0 but SN k lim xn – 0. k ; tÞSo it is important to note that weighted k-statistical convergence is neither a subset nor a superset of strongly ðN summable. Now the main definition is introduced as follows. Definition 2.2. Let / be a modulus function and ft n gn2N be a sequence of real numbers such that lim inf n!1 tn > 0 and P T abðnÞ ¼ k2½an ;bn t k , forall n 2 N. Then the sequence of random variables fX n gn2N is said to be weighted modulus ab-statistical convergence of order c (where 0 < c 6 1) in probability to a random variable X (where X : W!R) if for any ; d > 0,
lim
1
n!1 T
6 T abðnÞ : t k /ðPðjX k Xj P ÞÞ P dgj ¼ 0:
jfk c abðnÞ c
ðSab ;P / ;t n Þ
In this case, X n ! X and the class of all weighted modulus statistically convergent sequences of order c in probability is denoted by ðScab ; P / ; tn Þ. c
ðSa1b ;P/ ;t n Þ
c
ðSa2b ;P / ;t n Þ
Theorem 2.1. If X n ! X and X n ! Y then PfX ¼ Yg ¼ 1, for any c1 ; c2 . Proof. Without any loss of generality assume that c1 6 c2 . If possible let PfX ¼ Yg – 1. Then there exists two positive real numbers ; d such that /ðPðjX Yj P ÞÞ > d and lim inf n!1 t n > d. We know the following inequality
þ / P jX n Yj P ; for all n 2 N; /ðPðjX Yj P ÞÞ 6 / P jX n Xj P 2 2
(by using the condition of Definition 1.7 (ii) and the inequality PðjX Yj P Þ 6 P jX n Xj P 2 þ P jX n Yj P 2 ; 8n 2 NÞ. Then 1c
T abðnÞ2 ¼ 6
1
k 6 T abðnÞ : t k /ðPðjX Yj P ÞÞ P d2
c T a2bðnÞ
(
( )
)
d2
1
d2
P P k 6 T : t / P jX Xj P þ k 6 T : t / P jX Yj P
abðnÞ abðnÞ k k k k c 2 2 T ca2bðnÞ
2 2
T a1bðnÞ
1
506
S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509
which is impossible because the right hand side tends to zero as n!1 but left hand side tends to 1 as n!1. Hence the result. h The following example shows that there is a sequence fX n gn2N of random variables which is weighted ab-statistical convergence of order c to a random variable X but it is not weighted modulus ab-statistical convergence of order c. Example 2.3. Let the sequence of random variables fX n gn2N is defined by,
(
Xn 2
f1; 1g; f0; 1g;
with p:m:f PðX n ¼ 1Þ ¼ PðX n ¼ 0Þ; with p:m:f PðX n ¼ 0Þ ¼ 1
1 n2
if n ¼ m2 ;
PðX n ¼ 1Þ ¼ n12 ;
;
where m 2 N; if n – m2 ;
where m 2 N:
pffiffiffi Let 12 < c 6 1; tn ¼ 2n; an ¼ n; bn ¼ n2 ; 8n 2 N and /ðxÞ ¼ x; 8x 2 ½0; 1Þ; then T abðnÞ ¼ n4 þ n; 8n 2 N. For 0 < ; d < 1, we get
1 T cabðnÞ
jfk 6 T abðnÞ
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 n4 þ n 2n 2 : t k PðjX k 0j P Þ P dgj 6 4 6 2ð2c1Þ c 6 n4c n ðn þ nÞ
and
1 c jfk 6 T abðnÞ : t k /ðPðjX k 0j P ÞÞ P dgj P T a1bðnÞ : T cabðnÞ So fX n gn2N 2 ðScab ; P; tn Þ but not in ðScab ; P / ; tn Þ. Next example shows that there is a sequence fX n gn2N of random variables which is weighted modulus ab-statistical convergence of order c2 to a random variable X but it is not weighted modulus ab-statistical convergence of order c1 for 0 < c1 < c2 6 1. Example 2.4. Let rs be a rational number between c1 and c2 and a; b; c; d are four different real constants. We consider a sequence of random variables fX n gn2N is defined by,
(
Xn 2
fa; bg; fc; dg;
with probability
1 ; 2
s
if n ¼ ½mr ;
with p:m:f PðX n ¼ cÞ ¼ 1
1 n3
;
for any m 2 N;
PðX n ¼ dÞ ¼ n13 ;
s
if n – ½mr ;
for any m 2 N:
For 0 < < 12 minfja cj; jb cjjd cjg,
(
PðjX n cj P Þ ¼
s
1; 1 n3
if n ¼ ½mr ; s
if n – ½mr ;
;
for any m 2 N; for any m 2 N:
Taking tn ¼ n; an ¼ n; bn ¼ n2 ; 8n 2 N and /ðxÞ ¼ Now let d > 0, then
1 c T a2bðnÞ
k 6 T abðnÞ : t k /ðPðjX k cj P ÞÞ P d 6
pffiffiffi x; 8x 2 ½0; 1Þ, then T abðnÞ ¼ n2 ðn3 þ 1Þ, forall n 2 N.
M1 r n4ðc2 sÞ
and
1 M3 r jfk 6 T abðnÞ : t k /ðPðjX k cj P ÞÞ P dgj P M 2 n4ðsc1 Þ 4c ; c n T a1bðnÞ c
ðSa2b ;P/ ;t n Þ
c
where M 1 ; M 2 and M 3 are positive constants. So we have X n ! c but fX n gn2N is not in ðSa1b ; P/ ; t n Þ converges to c. c
c
Remark 2.2. In Theorem 2 [3], if c1 < c2 then m01 m02 (i.e., statistical convergence of order c1 implies statistical convergence of order c2 ) and this inclusion is strict for at least those c1 ; c2 for which there is a k 2 N such that c1 < 1k < c2 . But the above Example 2.4 if we choose t n ¼ 1; an ¼ 1; bn ¼ n; 8n 2 N and /ðxÞ ¼ x; 8x 2 ½0; 1Þ, then it can be easily shows that the c c 1 inequality m01 m02 is strict for any c1 < c2 (i.e., c1 ; c2 may satisfy the inequality kþ1 < c1 < c2 < 1k for any k 2 N). c
ðSa1b ;P/ ;t n Þ
Theorem 2.2. Let 0 < c1 6 c2 6 1 and g : R!R be a continuous function on R. If X n ! X and PðjXj P aÞ ¼ 0 for some posic
ðSa2b ;P / ;t n Þ
tive real number a, then gðX n Þ ! gðXÞ. Proof. Since g is uniformly continuous on ½a; a, then in this interval, for each
jgðxk Þ gðxÞj < if jxk xj < d and k 2 N: It follows that
> 0 there exists d such that
507
S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509
/ðPðjgðX k Þ gðXÞj P ÞÞ 6 /ðPðjX k Xj P dÞÞ if k 2 N: Then for g > 0,
1 1 jfk 6 T abðnÞ : t k /ðPðjgðX k Þ gðXÞj P ÞÞ P ggj 6 c1 jfk 6 T abðnÞ : tk /ðPðjX k Xj P dÞÞ P ggj: c T a2bðnÞ T abðnÞ Hence the result.
h c
c
ðSa1b ;P / ;t n Þ
ðSa2b ;P / ;t n Þ
Corollary 2.1. Let 0 < c1 6 c2 6 1; X n ! x and g : R!R is a continuous function, then gðX n Þ ! gðxÞ. Proof is straight forward, so omitted. Definition 2.3. Let ft n gn2N be a sequence of real numbers such that t 1 > 0 and T abðnÞ ¼
P
k2½an ;bn t k !1
as n!1 and / be a
modulus function. The sequence of random variables fX n gn2N is said to be weighted modulus ab-strong Cesáro convergence of order c (where 0 < c 6 1) in probability to a random variable X if for any
lim
X
1
n!1 T
tk /ðPðjX k c abðnÞ k2½an ;bn
> 0,
Xj P ÞÞ ¼ 0:
c
ðNab ;P / ;t n Þ
In this case, X n ! X and the class of all weighted modulus statistically convergent sequences of order a in probability
is denoted by ðN cab ; P/ ; t n Þ. In the following, the relationship between ðScab ; P / ; tn Þ and ðN cab ; P/ ; tn Þ is investigated.
c
n Theorem 2.3. If 0 < c1 6 c2 6 1; lim inf n!1 tn > 0; 0 < an 6 1; 8n 2 N and lim inf n!1 T abbðnÞ > 1, then ðN a1b ; P/ ; t n Þ c2 / ðSab ; P ; tn Þ.
c
ðNa1b ;P / ;t n Þ
Proof. Let X n ! X and ; d > 0. Then
1
X
tk /ðPðjX k c T a1bðnÞ k2½an ;b n
d jfk 6 T abðnÞ : t k /ðPðjX k Xj P ÞÞ P dgj; c T a2bðnÞ
Xj P ÞÞ P
(since bn P T abðnÞ for sufficiently large values of n). Hence the result follows.
h c
The following example shows that, the sequence of random variables fX n gn2N in ðSa2b ; P/ ; tn Þ converges to X but not in c ðN a1b ; P / ; t n Þ converges to X. Example 2.5. Let c 2 ð0; 1Þ; c < c1 6 1 and 0 < c2 6 2c and a sequence of random variables fX n gn2N is defined by,
(
Xn 2
f1; 1g; f0; 1g;
with probability
1 ; 2
1
if n ¼ ½mc ; for any m 2 N;
with p:m:f PðX n ¼ 0Þ ¼ 1 n14 ;
PðX n ¼ 1Þ ¼ n14 ;
1
if n – ½mc ;
for any m 2 N:
pffiffiffi Next let t n ¼ 2n; an ¼ n; bn ¼ n2 ; 8n 2 N and /ðxÞ ¼ x; 8x 2 ½0; 1Þ. Then T abðnÞ ¼ nðn3 þ 1Þ; 8n 2 N. For any 0 < < 1, we get
( PðjX n 0j P Þ ¼
1
1; 1 n3
if n ¼ ½m c ; for any m 2 N; ;
1
if n – ½m c ; for any m 2 N:
Now let 0 < d < 1,
1 M1 jfk 6 T abðnÞ : t k /ðPðjX k 0j P ÞÞ P dgj 6 4ðc cÞ c n 1 T a1bðnÞ and
X 1 ðn2c nc 1Þ 1 tk /ðPðjX k 0j P ÞÞ P P M 2 4ðc c Þ > M3 > 0; c c ðn4 þ nÞ 2 T a2bðnÞ k2½an ;bn n 2 2 where M 1 ; M 2 and M 3 are positive real numbers. Hence the result. Theorem 2.4. Let 0 < c1 6 c2 6 1; lim inf n!1 tn > 0 and ft n gn2N be a bounded sequence of real numbers such that c c lim supn!1 cb2n < 1. Then ðSa1b ; P/ ; t n Þ ðN a2b ; P/ ; t n Þ. T abðnÞ
508
S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509 c ðSa1b ;P / ;t n Þ
Proof. Let X n ! X and t n 6 M 1 ; 8n 2 N and lim supn!1 cb2n ¼ M2 , where M 1 and M 2 are positive real numbers. For any T abðnÞ
;
d > 0, setting H ¼ fk 6 T abðnÞ : tk /ðPðjX k Xj P ÞÞ P dg. Then
X
1
t k /ðPðjX k c T a2bðnÞ k2½an ;b n
Xj P ÞÞ ¼ 6
1
X
t k /ðPðjX k c T a2bðnÞ k2½an ;b \H n
Xj P ÞÞ þ
X
1
t k /ðPðjX k c T a2bðnÞ k2½an ;b \Hc n
Xj P ÞÞ
M1 M3 jfk 6 T abðnÞ : t k /ðPðjX k Xj P ÞÞ P dgj þ M1 ðM2 þ 1Þd: c T a1bðnÞ
where M 3 is a positive constant. Since d is arbitary, so the result follows.
h
The following example shows that the sequence of random variables fX n gn2N in ðN cab ; P/ ; t n Þ converges to X but it is not in converges to X.
c ðSab ; P/ ; t n Þ
Example 2.6. Let c 2 ð0; 1Þ;
( Xn 2
f1; 0g;
c 2
< c 6 c and a sequence of random variables fX n gn2N is defined by,
with p:m:f PðX n ¼ 1Þ ¼ n12 ; with p:m:f PðX n ¼ 0Þ ¼ 1
f0; 1g;
1 n8
1
PðX n ¼ 0Þ ¼ 1 n12 ; PðX n ¼ 1Þ ¼
;
1 n8
if n ¼ ½mc ; 1 c
if n – ½m ;
;
where m 2 N;
where m 2 N:
pffiffiffi Let tn ¼ 2n; an ¼ n; bn ¼ n ; 8n 2 N and /ðxÞ ¼ x; 8x 2 ½0; 1Þ then T abðnÞ ¼ n4 þ n; 8n 2 N. For 0 < ; d < 1, we get 2
1
X
T cabðnÞ k2½an ;b
n
2 t k /ðPðjX k 0j P ÞÞ 6 4 c þ nÞ ðn 6
( 2c
1
c
ðn n þ 1Þ þ
13
þ
1 23
þ ... þ
!)
1 ðn2 Þ
3
M ðwhere M is a positive constantÞ n2ð2ccÞ
and c
1
jfk T cabðnÞ
6 T abðnÞ : t k /ðPðjX k 0j P ÞÞ P dgj P
ðn4 þ nÞ 1 4ðccÞ c P n 2 ðn4 þ nÞ
So fX n gn2N 2 ðN cab ; P / ; tn Þ but not in ðScab ; P / ; t n Þ. Now we would like to introduce the definitions of weighted modulus Sab -convergence of order c in probability and weighted modulus N ab -convergence of order c in probability for a sequence of random variables as follows: Definition 2.4. Let / be a modulus function and ft n gn2N be a sequence of real numbers such that lim inf n!1 t n > 0, and P T abðnÞ ¼ k2½an ;bn t k , forall n 2 N. Then the sequence of random variables fX n gn2N is said to be weighted modulus Sab -convergence of order c in probability (where 0 < c 6 1) to a random variable X if for every ; d > 0,
lim
n!1 T
1
jfk c abðnÞ
2 IabðnÞ : tk /ðPðjX k Xj P ÞÞ P dgj ¼ 0; c
ðWSab ;P/ ;t n Þ
where IabðnÞ ¼ ðT ½aðnÞ ; T ½bðnÞ and ½x denotes the greatest integer not grater than x. In this case we write X n ! X. The class of all weighted modulus Sab -convergence sequences of order c in probability is denoted by ðWScab ; P / ; tn Þ. c
ðWSa1b ;P / ;t n Þ
c
ðWSa2b ;P / ;t n Þ
It is very obvious that if X n ! X and X n ! Y then PfX ¼ Yg ¼ 1 for any c1 ; c2 . P Definition 2.5. Let ftn gn2N be a sequence of real numbers such that t1 > 0 and T abðnÞ ¼ k2½an ;bn t k !1, as n!1 and / be a modulus function. The sequence of random variables fX n gn2N is said to be weighted modulus N ab -convergence of order c in probability (where 0 < c 6 1) to a random variable X if for any > 0,
lim
n!1 T
1
X
t k /ðPðjX k c abðnÞ k2IabðnÞ
Xj P ÞÞ ¼ 0:
c
ðWNab ;P/ ;t n Þ
In this case, X n ! X and the class of all weighted modulus N ab -convergence sequences of order c in probability is denoted by ðWNcab ; P/ ; t n Þ. In the following, the relationship between ðWScab ; P/ ; tn Þ and ðWNcab ; P/ ; tn Þ is investigated. c
c
Theorem 2.5. Let 0 < c1 6 c2 6 1 and lim inf n!1 tn > 0. Then ðWNa1b ; P / ; tn Þ ðWSa2b ; P/ ; t n Þ. This inclusion is strict for any c1 6 c2 .
509
S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509
Proof. First part of this theorem, let ; d > 0, then
X
X
t k /ðPðjX k Xj P ÞÞ ¼
k2IabðnÞ ;t k /ðPðjX k XjPÞÞPd
k2IabðnÞ
X
tk /ðPðjX k Xj P ÞÞ þ
t k /ðPðjX k Xj P ÞÞ
k2IabðnÞ ;t k /ðPðjX k XjPÞÞ
P djfk 2 IabðnÞ : t k /ðPðjX k Xj P ÞÞ P dgj: For the second part we will give an example. pffiffiffi Let t n ¼ n; an ¼ n!; bn ¼ ðn þ 1Þ!; 8n 2 N and /ðxÞ ¼ x; 8x 2 ½0; 1Þ and a sequence of random variables fX n gn2N be defined by,
8 c1 > < f1; 1g; with p:m:f PðX n ¼ 1Þ ¼ PðX n ¼ 1Þ; if n is the first ½ðT ½bðnÞ T ½aðnÞ Þ 2 X n 2 integer in the interval ðT ½aðnÞ ; T ½bðnÞ ; > : f0; 1g; with p:m:f PðX n ¼ 0Þ ¼ 1 n13 ; PðX n ¼ 1Þ ¼ n13 ; otherwise: For 0 < ; d < 1, we get
1 1 jfk 2 IabðnÞ : t k /ðPðjX k 0j P ÞÞ P dgj 6 c1 !0; as n!0: c T a1bðnÞ ðT ½bðnÞ T ½aðnÞ Þ 2 For next
1
X
c
T a1bðnÞ k2I
h t k /ðPðjX k 0j P ÞÞ P
abðnÞ
c1
ðT ½bðnÞ T ½aðnÞ Þ 2
inh
o c1 i ðT ½bðnÞ T ½aðnÞ Þ 2 þ 1
2ðT ½bðnÞ T ½aðnÞ Þc1
P
1 >0 3
(since 1 þ ð1 þ 1Þ þ ð1 þ 2Þ þ . . . þ ð1 þ mÞ 6 n þ ðn þ 1Þ þ ðn þ 1Þ þ . . . þ ðn þ mÞ; 8n; m 2 N). Hence the result.
h
Acknowledgments Thankful to respected Editor and Referees for his/her careful reading of the paper and several valuable suggestions which has improved the quality and presentation of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
H. Aktug˘lu, Korovkin type approximation theorems proved via-statistical convergence, J. Comput. Appl. Math. 259 (2014) 174–181. C. Belen, S.A. Mohiuddine, Generalized weighted statistical convergence and application, Appl. Math. Comput. 219 (2013) 9821–9826. S. Bhunia, P. Das, S. Pal, Restricting statistical convergence, Acta Math. Hungar. 134 (1–2) (2012) 153–161. R. Çolak, Statistical convergence of order a, Modern methods in Analy. Appl., Anamaya Pub, New Delhi, India, 2010 (pp. 121–129). R. Çolak, C.A. Bektasß, Statistical convergence of order a, Acta Math. Scientia 31B (3) (2011) 953–959. J.S. Connor, The statistical and strong p-Cesaro convergence sequence, Analysis 8 (1988) 47–63. P. Das, E. Savas, On I -statistical pre-Cauchy sequences, Taiwan J. Math. 18 (1) (2014) 115–125. P. Das, S. Ghosal, S. Som, Statistical convergence of order a in probability, Arab. J. Math. Sci. doi: 10.1016/j.ajmsc.2014.06.002. O. Edely, M. Mursaleen, A. Khan, Approximation for periodic functions via weighted statistical convergence, Appl. Math. Comput. 219 (2013) 8231– 8236. H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244 (in French). A.R. Freedman, J.J. Sember, M. Raphael, Some Cesaro-type summability space, Proc. Lond. Math. Soc. 37 (3) (1978) 508–520. J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301–313. J.A. Fridy, C. Orhan, Lacunary statistical convergence, Pacific J. Math. 160 (1993) 43–51. S. Ghosal, Statistical convergence of a sequence of random variables and limit theorems, Appl. Math. 58 (4) (2013) 423–437. S. Ghosal, I -Statistical convergence of a sequence of random variables in probability, Afrika Math. 25 (3) (2014) 681–692. S. Ghosal, Sk -Statistical convergence of a sequence of random variables, J. Egypt. Math. Soc. (2014), http://dx.doi.org/10.1016/ j.joems.03.007. S. Ghosal, Weighted statistical convergence of order a and its applications, J. Egypt. Math. Soc. doi:10.1016/j.joems.2014.08.006. M. Gürdal, On ideal convergent sequences in 2-normed spaces, Thai J. Math. 4 (1) (2006) 85–91. M. Gürdal, A. Sahiner, Ideal Convergence in n-normed spaces and some new sequence spaces via n-norm, J. Fundam. Sci. 4 (1) (2008) 233–244. M. Gürdal, H. Sari, Extremal a-statistical limit points via ideals, J. Egypt. Math. Soc. 22 (2014) 55–58. M. Gürdal, M.B. Huban, On I -convergence of double sequences in the topology induced by random 2-norms, Matematicki Vesnik 66 (1) (2014) 73–83. V. Karakaya, T.A. Chishti, Weighted statistical convergence, Iran. J. Sci. Techno. Trans. A 33 (A3) (2009) 219–223. I.J. Maddox, Sequence spaces define by a modulus, Math. Proc. Cambridge Philos. Soc. 100 (1986) 161–166. M. Mursaleen, k-Statistical convergence, Math. Slovaca 50 (1) (2000) 111–115. M. Mursaleen, V. Karakaya, M. Erturk, F. Gursoy, Weighted statistical convergence and its application to Korovkin type approximation theorem, Appl. Math. Comput. 218 (2012) 9132–9137. T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca 30 (1980) 139–150. E. Savasß, On some generalized sequence space defined by a modulus, Indian J. Pure. Appl. Math. 38 (1) (1999) 459–464. E. Savasß, On some new double lacunary sequences spaces via Orlicz function, J. Comput. Anal. Appl. 11 (3) (2009) 423–430. E. Savasß, R.F. Patterson, Some double lacunary sequence spaces defined by Orlicz functions, Southeast Asian Bull. Math. 35 (1) (2011) 103–110. I.J. Schonenberg, The integrability of certain functions and related summability methods, Am. Math. Mon. 66 (1959) 361–375. C. Sßençimen, Statistical convergence in probability for a sequence of random functions, J. Theor. Probab. 26 (2013) 94–106. H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 2 (1951) 73–74. A. Zygmund, Trigonometric Series, second ed., Cambridge Univ. Press, Cambridge, 1979.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Generalized weighted random convergence in probability Sanjoy Ghosal Department of Mathematics, Kalyani Government Engineering College, Nadia, Kalyani 741235, West Bengal, India
a r t i c l e
i n f o
Keywords: Weighted statistical convergence Weighted k-statistical convergence Weighted modulus ab-statistical convergence of order c Weighted modulus ab-strong Cesáro convergence of order c Weighted modulus Sab -convergence of order c Weighted modulus N ab -convergence of order c
a b s t r a c t The definition of weighted statistical convergence was first introduced by Karakaya & Chishti (2009) and later on Mursaleen et al. (2012) modified the definition of this concept. Using the modified definition, Edely et al. (2013) had researched further in approximation theory for periodic functions and Belen & Mohiuddine (2013) had further extended to the new-definition of weighted k-statistical convergence. But some problems are still there, so the definition of weighted k-statistical convergence is needed to modify. In this paper, we will introduce some new constraints which will make the definition of weighted k-statistical convergence is more useful by using the definition of ab-statistical convergence. Using it and independently, some newly developed concepts of the convergence of a sequence of random variables in probability, namely, weighted modulus ab-statistical convergence of order c, weighted modulus ab-strong Cesáro convergence of order c, weighted modulus Sab -convergence of order c, and weighted modulus Nab -convergence of order c, have been introduced and their basic interrelations also have been investigated. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The concept of statistical convergence was introduced by Fast [10] and Steinhaus [32] and later on reintroduced by Schonberg [30] independently and is based on the notion of asymptotic density of the subset of natural numbers. However, the first idea of statistical convergence (by different name) was given by Zygmund [33] in the first edition of his monograph published in Warsaw in 1935. Later on it was further investigated from the sequence space point of view and linked with summability theorem by Fridy [12], Connor [6], Šalát [26], Gürdal [18], Das & Savas [7], Mursaleen [24], Fridy and Orhan [13]. In last five years, it has several devolvement and application of this convergence has been given by many author’s like: (i) statistical convergence of order a by Çolak [4] (statistical convergence of order a was also independently introduce by Bhunia et. al. [3]), (ii) k-statistical convergence of order a by Çolak and Bektasß [5], (iii) weighted statistical convergence by Mursaleen et. al. [25], (iv) weighted k-statistical convergence by Belen and Mohiuddine [2], (v) statistical convergence in probability by Ghosal [14,15], (vi) statistical convergence of a sequence of random function in probabilistic metric space by Sencimen [31], (vii) ab-statistical convergence by Aktug˘lu [1], (viii) A-statistical limit points via ideals by Gürdal and Sari [20], (ix) ideal convergence in n-norms space by Gürdal [19] and Gürdal and Huban [21], and so on. E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2014.10.056 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509
503
The entire mathematical theory of probability was built by three objects, namely (a) the event space W, (b) the class of events M and (c) the probability function P : M!R. The order 3-tuples ðW; M; PÞ is called probability space. A mapping X of W to R is called a random variable or stochastic variable or a variate if for any x 2 R, the set
fx : 1 < XðxÞ 6 x; x 2 Wg 2 M: Ordinarily, the distribution of a random variable is required to obtain probability connected to the random variable. But Tchebycheff’s inequality indicates that, irrespective of the shape of the density curve
Pðm < X < m þ Þ > 1
r2 hold; 2
(where X has finite mean m and variance r), i.e., the probability that X takes values in the interval ðm ; m þ Þ centred at m is closed to 1, provided r is sufficiently small, i.e., we get and upper bound for the probability that the deviation of X from its mean is at least units in terms of variance of X and . No assumption on the distribution of X. From the practical point of view the discussion of a random variable X is highly significant if it is known that there exists a real constant x for which PðjX xj < Þ ’ 1, where > 0 is sufficiently small, that is, it is nearly certain that values of X lie in a very small neighbourhood of x. For a sequence of random variables fX n gn2N , each X n may not have the above property but it may happen that the aforesaid property (with respect to a real constant x) becomes more and more distinguishable as n gradually increases and the question of existence of such a real constant x can be answered by a concept of convergence in probability of the sequence fX n gn2N . In this paper the ideas of four types of convergences of a sequence of random variables, namely, (a) weighted modulus ab-statistical convergence of order c, (b) weighted modulus ab-strong Cesáro convergence of order c, (c) weighted modulus Sab -convergence of order c, (d) weighted modulus N ab -convergence of order c, have been introduced and the interrelations among them have been investigated. Also their certain basic properties have been studied. The main object of this paper is to modify the definition of weighted k-statistical convergence and to establish some important theorems related to the modes of convergences (a) to (d), which will effectively extend and improve all the existing results in this direction ([1–6,8–17,22–27]). Moreover, intend to establish the relations among these four summability notions. It is important to note that the methods of proofs and in particular the examples are not analogous to the real case. The following definitions and notions will be needed in sequel. Definition 1.1 (see [3,4]). A sequence fxn gn2N of real numbers is said to be statistically convergent of order c (where 0 < c 6 1) to x if for every > 0,
lim
1
n!1 nc
jfk 6 n : jxk xj P gj ¼ 0: Sc
In this case we write xn ! x and the set of all statistically convergent sequences of order c is denoted by Sc . Definition 1.2 (see [5]). Let fkn gn2N be a sequence of non-decreasing sequence of positive real numbers tending to 1 such that knþ1 6 kn þ 1; k1 ¼ 1. The sequence of real numbers fxn gn2N is said to be k-statistically convergent of order c (where 0 < c 6 1) to x if for any > 0,
lim
1 c
n!1 k n
jfk 2 In : jxk xj P gj ¼ 0; c
Sk
where In ¼ ½n kn þ 1; n. In this case we write xn ! x and the set of all k-statistically convergent sequences of order c is denoted by Sck . P Definition 1.3 (see [25]). Let ft n gn2N be a sequence of nonnegative real numbers such that t1 > 0 and T n ¼ nk¼1 t n !1 as n!1. Then the sequence of real numbers fxn gn2N is said to be weighted statistically convergent (or, SN -convergent) to x if for every > 0,
lim
1
n!1 T n
jfk 6 T n : tk jxk xj P gj ¼ 0: SN
In this case we write xn ! x and the set of all weighted statistically convergent sequences is denoted by SN .
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S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509
Definition 1.4 (see [2]). Let ft n gn2N be a sequence of nonnegative real numbers such that t1 > 0 and T kn ¼ k ; tÞ-summable to x if n!1. A sequence of real numbers fxn gn2N is said to be strongly ðN
P
k2In t k !1,
as
1 X t jx xj ¼ 0; k2In k k n!1 T k n lim
;t ½N k
where In ¼ ½n kn þ 1; n. In this case we write xn ! x and the set of all weighted k-statistically convergent sequences is k ; t. denoted by ½N Definition 1.5 (see [14]). Let fX n gn2N be a sequence of random variables, where each X n is defined on the same sample space W (for each n) with respect to a given class of events M and a given probability function P : M!R. The sequence fX n gn2N is said to be statistically convergent in probability to a random variable X (where X : W!R) if for any ; d > 0,
lim
1
n!1 n
jfk 6 n : PðjX k Xj P Þ P dgj ¼ 0: ðS;PÞ
In this case we write X n ! X and the class of all statistically convergent sequences of random variables in probability is denoted by ðS; PÞ. For more results on this convergence see the paper [8]. Definition 1.6 (see [1]). Let fan gn2N and fbn gn2N be two sequences of positive real numbers such that (i) a and b are both non-decreasing, (ii) bn P an ; 8n 2 N, (iii) bn an !1 as n!1. Then the sequence of real numbers fxn gn2N is said to be ab-statistically convergent of order c (where 0 < c 6 1) to x if for every > 0,
lim
n!1 ðb n
1 jfk 2 ½an ; bn : jxk xj P gj ¼ 0: an þ 1Þc c
Sab
In this case we write xn ! x and the set of all weighted statistically convergent sequences of order c is denoted by Scab . Definition 1.7 (see [23,27]). A modulus function / is a function from ½0; 1Þ to ½0; 1Þ such that (i) /ðxÞ ¼ 0 if and only if x ¼ 0, (ii) /ðx þ yÞ 6 /ðxÞ þ /ðyÞ, forall x; y > 0, (iii) / is increasing, (iv) / is continuous from the right at zero. A modulus function may be bounded or unbounded. Maddox [23], Savasß [27,28], Savasß and Patterson [29] and other authors used modulus function to construct new sequence spaces. 2. Main results Belen and Mohiuddine [2] first defined the concept of weighted k-statistical convergence as follows: Let ft n gn2N be a P sequence of nonnegative real numbers such that t1 > 0 and T kn ¼ k2In tk !1, as n!1 (where In ¼ ½n kn þ 1; n). Then the sequence of real numbers fxn gn2N is said to be weighted k-statistically convergent (or, SN k -convergent) to x if for every > 0,
lim
1
n!1 T k
jfk 6 T kn : t k jxk xj P gj ¼ 0:
n
SN
k
In this case we write xn ! x (or SN k lim xn ¼ x) and the set of all weighted k-statistically convergent sequences is denoted by SN k . Remark 2.4 (iv) (from [2]) shows that, if kn ¼ n for all n 2 N, then SN k -convergence coincides with SN -convergence. But the definition of weighted k-statistical convergence is not well defined in general. This follows from the following example. Example 2.1. Let tn ¼ n1p (where 0 < p 1), kn ¼ n, forall n 2 N and a sequence of real numbers fxn gn2N is defined by,
xn ¼
0; if n ¼ 2m; for any m 2 N; 1;
otherwise: SN
k
SN
It is quite clear that xn ! x (or, xn ! x) where x be any real number, i.e., for any bounded real sequence fxn gn2N is weighted k-statistically convergent (or weighted statistical convergence) to any real number. Hence the definition is not well defined. So the definition of weighted k-statistical convergence need to be modified. Now, we are going to modify the definition as follows:
S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509
505
P Definition 2.1. Let ftn gn2N be a sequence of real numbers such that lim inf n!1 tn > 0 and T abðnÞ ¼ k2½an ;bn tk , forall n 2 N. Then the sequence of real numbers fxn gn2N is said to be weighted ab-statistically convergent of order c (where 0 < c 6 1) to x if for every > 0,
1
lim
n!1 T
6 T abðnÞ : t k jxk xj P gj ¼ 0:
jfk c abðnÞ
c
ðSab ;t n Þ
In this case we write xn ! x. The class of all weighted ab-statistical convergence sequences of order c is denoted by ðScab ; tn Þ. Remark 2.1 (i) For c ¼ 1; an ¼ 1 and bn ¼ n, then weighted ab-statistical convergence of order c coincides with weighted statistical convergence. (ii) For c ¼ 1; an ¼ n kn þ 1 and bn ¼ n, then weighted ab-statistical convergence of order c coincides with weighted k-statistical convergence and so on. c
c
ðSab ;t n Þ
ðSab ;t n Þ
It is obvious that if xn ! x and xn ! y then x ¼ y. k ; tÞ-summable to x then it is weighted In Theorem 2.7 (i) (from [2]), say that ‘‘If a sequence fxn gn2N is strongly ðN k-statistical convergent to x.’’ Which is not true given by the following example. Example 2.2. Let tn ¼ n; kn ¼ n, where n 2 N and a sequence of real numbers fxn gn2N is defined by,
(
xn ¼
1 ; n
if n is an even integer; otherwise:
0;
Now let 0 < < 1. Then for the large value of n we get, Tn 1X 2 1 1 and t k jxk 0j 6 jfk 6 T n : t k jxk 0j P gj P 3 P : T n k2I nþ1 Tn 3 Tn n
k ;t ½N
So xn ! 0 but SN k lim xn – 0. k ; tÞSo it is important to note that weighted k-statistical convergence is neither a subset nor a superset of strongly ðN summable. Now the main definition is introduced as follows. Definition 2.2. Let / be a modulus function and ft n gn2N be a sequence of real numbers such that lim inf n!1 tn > 0 and P T abðnÞ ¼ k2½an ;bn t k , forall n 2 N. Then the sequence of random variables fX n gn2N is said to be weighted modulus ab-statistical convergence of order c (where 0 < c 6 1) in probability to a random variable X (where X : W!R) if for any ; d > 0,
lim
1
n!1 T
6 T abðnÞ : t k /ðPðjX k Xj P ÞÞ P dgj ¼ 0:
jfk c abðnÞ c
ðSab ;P / ;t n Þ
In this case, X n ! X and the class of all weighted modulus statistically convergent sequences of order c in probability is denoted by ðScab ; P / ; tn Þ. c
ðSa1b ;P/ ;t n Þ
c
ðSa2b ;P / ;t n Þ
Theorem 2.1. If X n ! X and X n ! Y then PfX ¼ Yg ¼ 1, for any c1 ; c2 . Proof. Without any loss of generality assume that c1 6 c2 . If possible let PfX ¼ Yg – 1. Then there exists two positive real numbers ; d such that /ðPðjX Yj P ÞÞ > d and lim inf n!1 t n > d. We know the following inequality
þ / P jX n Yj P ; for all n 2 N; /ðPðjX Yj P ÞÞ 6 / P jX n Xj P 2 2
(by using the condition of Definition 1.7 (ii) and the inequality PðjX Yj P Þ 6 P jX n Xj P 2 þ P jX n Yj P 2 ; 8n 2 NÞ. Then 1c
T abðnÞ2 ¼ 6
1
k 6 T abðnÞ : t k /ðPðjX Yj P ÞÞ P d2
c T a2bðnÞ
(
( )
)
d2
1
d2
P P k 6 T : t / P jX Xj P þ k 6 T : t / P jX Yj P
abðnÞ abðnÞ k k k k c 2 2 T ca2bðnÞ
2 2
T a1bðnÞ
1
506
S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509
which is impossible because the right hand side tends to zero as n!1 but left hand side tends to 1 as n!1. Hence the result. h The following example shows that there is a sequence fX n gn2N of random variables which is weighted ab-statistical convergence of order c to a random variable X but it is not weighted modulus ab-statistical convergence of order c. Example 2.3. Let the sequence of random variables fX n gn2N is defined by,
(
Xn 2
f1; 1g; f0; 1g;
with p:m:f PðX n ¼ 1Þ ¼ PðX n ¼ 0Þ; with p:m:f PðX n ¼ 0Þ ¼ 1
1 n2
if n ¼ m2 ;
PðX n ¼ 1Þ ¼ n12 ;
;
where m 2 N; if n – m2 ;
where m 2 N:
pffiffiffi Let 12 < c 6 1; tn ¼ 2n; an ¼ n; bn ¼ n2 ; 8n 2 N and /ðxÞ ¼ x; 8x 2 ½0; 1Þ; then T abðnÞ ¼ n4 þ n; 8n 2 N. For 0 < ; d < 1, we get
1 T cabðnÞ
jfk 6 T abðnÞ
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 n4 þ n 2n 2 : t k PðjX k 0j P Þ P dgj 6 4 6 2ð2c1Þ c 6 n4c n ðn þ nÞ
and
1 c jfk 6 T abðnÞ : t k /ðPðjX k 0j P ÞÞ P dgj P T a1bðnÞ : T cabðnÞ So fX n gn2N 2 ðScab ; P; tn Þ but not in ðScab ; P / ; tn Þ. Next example shows that there is a sequence fX n gn2N of random variables which is weighted modulus ab-statistical convergence of order c2 to a random variable X but it is not weighted modulus ab-statistical convergence of order c1 for 0 < c1 < c2 6 1. Example 2.4. Let rs be a rational number between c1 and c2 and a; b; c; d are four different real constants. We consider a sequence of random variables fX n gn2N is defined by,
(
Xn 2
fa; bg; fc; dg;
with probability
1 ; 2
s
if n ¼ ½mr ;
with p:m:f PðX n ¼ cÞ ¼ 1
1 n3
;
for any m 2 N;
PðX n ¼ dÞ ¼ n13 ;
s
if n – ½mr ;
for any m 2 N:
For 0 < < 12 minfja cj; jb cjjd cjg,
(
PðjX n cj P Þ ¼
s
1; 1 n3
if n ¼ ½mr ; s
if n – ½mr ;
;
for any m 2 N; for any m 2 N:
Taking tn ¼ n; an ¼ n; bn ¼ n2 ; 8n 2 N and /ðxÞ ¼ Now let d > 0, then
1 c T a2bðnÞ
k 6 T abðnÞ : t k /ðPðjX k cj P ÞÞ P d 6
pffiffiffi x; 8x 2 ½0; 1Þ, then T abðnÞ ¼ n2 ðn3 þ 1Þ, forall n 2 N.
M1 r n4ðc2 sÞ
and
1 M3 r jfk 6 T abðnÞ : t k /ðPðjX k cj P ÞÞ P dgj P M 2 n4ðsc1 Þ 4c ; c n T a1bðnÞ c
ðSa2b ;P/ ;t n Þ
c
where M 1 ; M 2 and M 3 are positive constants. So we have X n ! c but fX n gn2N is not in ðSa1b ; P/ ; t n Þ converges to c. c
c
Remark 2.2. In Theorem 2 [3], if c1 < c2 then m01 m02 (i.e., statistical convergence of order c1 implies statistical convergence of order c2 ) and this inclusion is strict for at least those c1 ; c2 for which there is a k 2 N such that c1 < 1k < c2 . But the above Example 2.4 if we choose t n ¼ 1; an ¼ 1; bn ¼ n; 8n 2 N and /ðxÞ ¼ x; 8x 2 ½0; 1Þ, then it can be easily shows that the c c 1 inequality m01 m02 is strict for any c1 < c2 (i.e., c1 ; c2 may satisfy the inequality kþ1 < c1 < c2 < 1k for any k 2 N). c
ðSa1b ;P/ ;t n Þ
Theorem 2.2. Let 0 < c1 6 c2 6 1 and g : R!R be a continuous function on R. If X n ! X and PðjXj P aÞ ¼ 0 for some posic
ðSa2b ;P / ;t n Þ
tive real number a, then gðX n Þ ! gðXÞ. Proof. Since g is uniformly continuous on ½a; a, then in this interval, for each
jgðxk Þ gðxÞj < if jxk xj < d and k 2 N: It follows that
> 0 there exists d such that
507
S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509
/ðPðjgðX k Þ gðXÞj P ÞÞ 6 /ðPðjX k Xj P dÞÞ if k 2 N: Then for g > 0,
1 1 jfk 6 T abðnÞ : t k /ðPðjgðX k Þ gðXÞj P ÞÞ P ggj 6 c1 jfk 6 T abðnÞ : tk /ðPðjX k Xj P dÞÞ P ggj: c T a2bðnÞ T abðnÞ Hence the result.
h c
c
ðSa1b ;P / ;t n Þ
ðSa2b ;P / ;t n Þ
Corollary 2.1. Let 0 < c1 6 c2 6 1; X n ! x and g : R!R is a continuous function, then gðX n Þ ! gðxÞ. Proof is straight forward, so omitted. Definition 2.3. Let ft n gn2N be a sequence of real numbers such that t 1 > 0 and T abðnÞ ¼
P
k2½an ;bn t k !1
as n!1 and / be a
modulus function. The sequence of random variables fX n gn2N is said to be weighted modulus ab-strong Cesáro convergence of order c (where 0 < c 6 1) in probability to a random variable X if for any
lim
X
1
n!1 T
tk /ðPðjX k c abðnÞ k2½an ;bn
> 0,
Xj P ÞÞ ¼ 0:
c
ðNab ;P / ;t n Þ
In this case, X n ! X and the class of all weighted modulus statistically convergent sequences of order a in probability
is denoted by ðN cab ; P/ ; t n Þ. In the following, the relationship between ðScab ; P / ; tn Þ and ðN cab ; P/ ; tn Þ is investigated.
c
n Theorem 2.3. If 0 < c1 6 c2 6 1; lim inf n!1 tn > 0; 0 < an 6 1; 8n 2 N and lim inf n!1 T abbðnÞ > 1, then ðN a1b ; P/ ; t n Þ c2 / ðSab ; P ; tn Þ.
c
ðNa1b ;P / ;t n Þ
Proof. Let X n ! X and ; d > 0. Then
1
X
tk /ðPðjX k c T a1bðnÞ k2½an ;b n
d jfk 6 T abðnÞ : t k /ðPðjX k Xj P ÞÞ P dgj; c T a2bðnÞ
Xj P ÞÞ P
(since bn P T abðnÞ for sufficiently large values of n). Hence the result follows.
h c
The following example shows that, the sequence of random variables fX n gn2N in ðSa2b ; P/ ; tn Þ converges to X but not in c ðN a1b ; P / ; t n Þ converges to X. Example 2.5. Let c 2 ð0; 1Þ; c < c1 6 1 and 0 < c2 6 2c and a sequence of random variables fX n gn2N is defined by,
(
Xn 2
f1; 1g; f0; 1g;
with probability
1 ; 2
1
if n ¼ ½mc ; for any m 2 N;
with p:m:f PðX n ¼ 0Þ ¼ 1 n14 ;
PðX n ¼ 1Þ ¼ n14 ;
1
if n – ½mc ;
for any m 2 N:
pffiffiffi Next let t n ¼ 2n; an ¼ n; bn ¼ n2 ; 8n 2 N and /ðxÞ ¼ x; 8x 2 ½0; 1Þ. Then T abðnÞ ¼ nðn3 þ 1Þ; 8n 2 N. For any 0 < < 1, we get
( PðjX n 0j P Þ ¼
1
1; 1 n3
if n ¼ ½m c ; for any m 2 N; ;
1
if n – ½m c ; for any m 2 N:
Now let 0 < d < 1,
1 M1 jfk 6 T abðnÞ : t k /ðPðjX k 0j P ÞÞ P dgj 6 4ðc cÞ c n 1 T a1bðnÞ and
X 1 ðn2c nc 1Þ 1 tk /ðPðjX k 0j P ÞÞ P P M 2 4ðc c Þ > M3 > 0; c c ðn4 þ nÞ 2 T a2bðnÞ k2½an ;bn n 2 2 where M 1 ; M 2 and M 3 are positive real numbers. Hence the result. Theorem 2.4. Let 0 < c1 6 c2 6 1; lim inf n!1 tn > 0 and ft n gn2N be a bounded sequence of real numbers such that c c lim supn!1 cb2n < 1. Then ðSa1b ; P/ ; t n Þ ðN a2b ; P/ ; t n Þ. T abðnÞ
508
S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509 c ðSa1b ;P / ;t n Þ
Proof. Let X n ! X and t n 6 M 1 ; 8n 2 N and lim supn!1 cb2n ¼ M2 , where M 1 and M 2 are positive real numbers. For any T abðnÞ
;
d > 0, setting H ¼ fk 6 T abðnÞ : tk /ðPðjX k Xj P ÞÞ P dg. Then
X
1
t k /ðPðjX k c T a2bðnÞ k2½an ;b n
Xj P ÞÞ ¼ 6
1
X
t k /ðPðjX k c T a2bðnÞ k2½an ;b \H n
Xj P ÞÞ þ
X
1
t k /ðPðjX k c T a2bðnÞ k2½an ;b \Hc n
Xj P ÞÞ
M1 M3 jfk 6 T abðnÞ : t k /ðPðjX k Xj P ÞÞ P dgj þ M1 ðM2 þ 1Þd: c T a1bðnÞ
where M 3 is a positive constant. Since d is arbitary, so the result follows.
h
The following example shows that the sequence of random variables fX n gn2N in ðN cab ; P/ ; t n Þ converges to X but it is not in converges to X.
c ðSab ; P/ ; t n Þ
Example 2.6. Let c 2 ð0; 1Þ;
( Xn 2
f1; 0g;
c 2
< c 6 c and a sequence of random variables fX n gn2N is defined by,
with p:m:f PðX n ¼ 1Þ ¼ n12 ; with p:m:f PðX n ¼ 0Þ ¼ 1
f0; 1g;
1 n8
1
PðX n ¼ 0Þ ¼ 1 n12 ; PðX n ¼ 1Þ ¼
;
1 n8
if n ¼ ½mc ; 1 c
if n – ½m ;
;
where m 2 N;
where m 2 N:
pffiffiffi Let tn ¼ 2n; an ¼ n; bn ¼ n ; 8n 2 N and /ðxÞ ¼ x; 8x 2 ½0; 1Þ then T abðnÞ ¼ n4 þ n; 8n 2 N. For 0 < ; d < 1, we get 2
1
X
T cabðnÞ k2½an ;b
n
2 t k /ðPðjX k 0j P ÞÞ 6 4 c þ nÞ ðn 6
( 2c
1
c
ðn n þ 1Þ þ
13
þ
1 23
þ ... þ
!)
1 ðn2 Þ
3
M ðwhere M is a positive constantÞ n2ð2ccÞ
and c
1
jfk T cabðnÞ
6 T abðnÞ : t k /ðPðjX k 0j P ÞÞ P dgj P
ðn4 þ nÞ 1 4ðccÞ c P n 2 ðn4 þ nÞ
So fX n gn2N 2 ðN cab ; P / ; tn Þ but not in ðScab ; P / ; t n Þ. Now we would like to introduce the definitions of weighted modulus Sab -convergence of order c in probability and weighted modulus N ab -convergence of order c in probability for a sequence of random variables as follows: Definition 2.4. Let / be a modulus function and ft n gn2N be a sequence of real numbers such that lim inf n!1 t n > 0, and P T abðnÞ ¼ k2½an ;bn t k , forall n 2 N. Then the sequence of random variables fX n gn2N is said to be weighted modulus Sab -convergence of order c in probability (where 0 < c 6 1) to a random variable X if for every ; d > 0,
lim
n!1 T
1
jfk c abðnÞ
2 IabðnÞ : tk /ðPðjX k Xj P ÞÞ P dgj ¼ 0; c
ðWSab ;P/ ;t n Þ
where IabðnÞ ¼ ðT ½aðnÞ ; T ½bðnÞ and ½x denotes the greatest integer not grater than x. In this case we write X n ! X. The class of all weighted modulus Sab -convergence sequences of order c in probability is denoted by ðWScab ; P / ; tn Þ. c
ðWSa1b ;P / ;t n Þ
c
ðWSa2b ;P / ;t n Þ
It is very obvious that if X n ! X and X n ! Y then PfX ¼ Yg ¼ 1 for any c1 ; c2 . P Definition 2.5. Let ftn gn2N be a sequence of real numbers such that t1 > 0 and T abðnÞ ¼ k2½an ;bn t k !1, as n!1 and / be a modulus function. The sequence of random variables fX n gn2N is said to be weighted modulus N ab -convergence of order c in probability (where 0 < c 6 1) to a random variable X if for any > 0,
lim
n!1 T
1
X
t k /ðPðjX k c abðnÞ k2IabðnÞ
Xj P ÞÞ ¼ 0:
c
ðWNab ;P/ ;t n Þ
In this case, X n ! X and the class of all weighted modulus N ab -convergence sequences of order c in probability is denoted by ðWNcab ; P/ ; t n Þ. In the following, the relationship between ðWScab ; P/ ; tn Þ and ðWNcab ; P/ ; tn Þ is investigated. c
c
Theorem 2.5. Let 0 < c1 6 c2 6 1 and lim inf n!1 tn > 0. Then ðWNa1b ; P / ; tn Þ ðWSa2b ; P/ ; t n Þ. This inclusion is strict for any c1 6 c2 .
509
S. Ghosal / Applied Mathematics and Computation 249 (2014) 502–509
Proof. First part of this theorem, let ; d > 0, then
X
X
t k /ðPðjX k Xj P ÞÞ ¼
k2IabðnÞ ;t k /ðPðjX k XjPÞÞPd
k2IabðnÞ
X
tk /ðPðjX k Xj P ÞÞ þ
t k /ðPðjX k Xj P ÞÞ
k2IabðnÞ ;t k /ðPðjX k XjPÞÞ
P djfk 2 IabðnÞ : t k /ðPðjX k Xj P ÞÞ P dgj: For the second part we will give an example. pffiffiffi Let t n ¼ n; an ¼ n!; bn ¼ ðn þ 1Þ!; 8n 2 N and /ðxÞ ¼ x; 8x 2 ½0; 1Þ and a sequence of random variables fX n gn2N be defined by,
8 c1 > < f1; 1g; with p:m:f PðX n ¼ 1Þ ¼ PðX n ¼ 1Þ; if n is the first ½ðT ½bðnÞ T ½aðnÞ Þ 2 X n 2 integer in the interval ðT ½aðnÞ ; T ½bðnÞ ; > : f0; 1g; with p:m:f PðX n ¼ 0Þ ¼ 1 n13 ; PðX n ¼ 1Þ ¼ n13 ; otherwise: For 0 < ; d < 1, we get
1 1 jfk 2 IabðnÞ : t k /ðPðjX k 0j P ÞÞ P dgj 6 c1 !0; as n!0: c T a1bðnÞ ðT ½bðnÞ T ½aðnÞ Þ 2 For next
1
X
c
T a1bðnÞ k2I
h t k /ðPðjX k 0j P ÞÞ P
abðnÞ
c1
ðT ½bðnÞ T ½aðnÞ Þ 2
inh
o c1 i ðT ½bðnÞ T ½aðnÞ Þ 2 þ 1
2ðT ½bðnÞ T ½aðnÞ Þc1
P
1 >0 3
(since 1 þ ð1 þ 1Þ þ ð1 þ 2Þ þ . . . þ ð1 þ mÞ 6 n þ ðn þ 1Þ þ ðn þ 1Þ þ . . . þ ðn þ mÞ; 8n; m 2 N). Hence the result.
h
Acknowledgments Thankful to respected Editor and Referees for his/her careful reading of the paper and several valuable suggestions which has improved the quality and presentation of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
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