IN PROBABILITY. Kragujevac Journal of Mathematics Volume 42(1) (2018), Pages

Transcription

1 Kragujevac Journal of Mathematics Volume 4) 08), Pages A NOTION OF -STATISTICAL CONVERGENCE OF ORDER γ IN PROBABILITY PRATULANANDA DAS, SUMIT SOM, SANJOY GHOSAL, AND VATAN KARAKAYA 3 Abstract. A sequence of real numbers x n } n N is said to be -statistically convergent of order γ where 0 < γ ) to a real number x [] if for every δ > 0, β n α n + ) γ k [α n, β n ] : x k x δ} = 0, where α n } n N β n } n N are two sequences of positive real numbers such that α n } n N β n } n N are both non-decreasing, β n α n for all n N, β n α n ) as n. In this paper we study a related concept of convergences in which the value x k is replaced by P X k X ε) E X k X r ) respectively where X, X k are rom variables for each k N, ε > 0, P denotes the probability, E denotes the expectation) we call them -statistical convergence of order γ in probability -statistical convergence of order γ in r th expectation respectively. The results are applied to build the probability distribution for -strong p-cesàro summability of order γ in probability -statistical convergence of order γ in distribution. So our main objective is to interpret a relational behaviour of above mentioned four convergences. We give a condition under which a sequence of rom variables will converge to a unique it under two different α, β) sequences this is also use to prove that if this condition violates then the it value of statistical convergence of order γ in probability of a sequence of rom variables for two different α, β) sequences may not be equal.. Introduction The idea of convergence of a real sequence has been extended to statistical convergence by Fast [9] Steinhaus [9] later on reintroduced by Schoenberg Key words phrases. -statistical convergence, -statistical convergence of order γ in probability, -strong p-cesàro summability of order γ in probability, -statistical convergence of order γ in r th expectation, -statistical convergence of order γ in distribution. 00 Mathematics Subject Classification. Primary: 40A35. Secondary: 40G5, 60B0. Received: July 9, 06. Accepted: November 4, 06. 5

2 5 P. DAS, S. SOM, S. GHOSAL, AND V. KARAKAYA [7] independently 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 [0] in the first edition of his monograph published in Warsaw in 935. Later on it was further investigated from the sequence space point of view linked with summability theorem by Fridy [], Connor [4], Šalát [6], Das et al. [6], Fridy Orhan []. In [, 3] a different direction was given to the study of statistical convergence where the notion of statistical convergence of order γ 0 < γ < ) was introduced by using the notion of natural density of order γ where n is replaced by n γ in the denominator in the definition of natural density). It was observed in [], that the behavior of this new convergence was not exactly parallel to that of statistical convergence some basic properties were obtained. More results on this convergence can be seen from [8]. In this context it should be noted that the history of strong p-cesàro summability is not so clear. Connor in [4], observed that if a sequence is strongly p-cesàro summable of order γ for 0 < p < ) to x, then the sequence must be statistically convergent of order γ to the same it. Both Fast [9] Schoenberg [7] noted that if a bounded sequence is statistically convergent to x, then it is strongly Cesàro summable to x. But in the more general case of order γ the result may not be true, as was established in [3]. In [0], the relation between strongly Cesàro summable N θ -convergence was established among other things. Recently the idea of statistical convergence of order γ was further extended to -statistical convergence of order γ in [] as follows: let α = α n } n N, β = β n } n N be two non-decreasing sequences of positive real numbers satisfying the conditions α n β n for all n N β n α n ) as n. This pair of sequence we denoted by α, β). Then a sequence x n } n N of real numbers is said to be -statistically convergent of order γ where 0 < γ ) to a real number x if for each ε > 0, the set K = n N : x n x ε} has -natural density zero, i.e. β n α n + ) k [α n, β γ n ] : x k x ε} = 0 we write S γ x S γ n = x or x n x. -statistical convergence of order γ is more general than statistical convergence of order γ, lacunary statistical convergence of order γ λ statistical convergence of order γ if we take i) α n = β n = n, for all n N, ii) α r = k r + ) β r = k r for all r N where k r } r N 0} is a lacunary sequence, iii) α n = n λ n + ) β n = n, for all n N respectively. On the other h in probability theory, a new type of convergence called statistical convergence in probability was introduced in [4] as follows: let X n } n N be a sequence of rom variables where each X n is defined on the same sample space S for each n) with respect to a given class of events a given probability function P : R. Then the sequence X n } n N is said to be statistically convergent in probability to a

3 A NOTION OF -STATISTICAL CONVERGENCE OF ORDER γ IN PROBABILITY 53 rom variable X where X : S R) if for any ε, δ > 0 n k n : P X k X ε) δ} = 0. P S In this case we write X n X. The class of all sequences of rom variables which are statistically convergent in probability is denoted by P S. One can also see [7,8,5] for related works. In a natural way, in this paper we combine the approaches of the above mentioned papers introduce new more general methods, namely, -statistical convergence of order γ in probability, -strong p-cesàro summability of order γ in probability, -statistical convergence of order γ in r th expectation -statistical convergence of order γ in distribution. We mainly investigate their relationship also make some observations about these classes. In the way we show that -statistical it of order γ 0 < γ < ) of a sequence of rom variables for two different α, β) sequences may not be equal. It is important to note that the method of proofs in particular examples are not analogous to the real case.. -Statistical Convergence of Order γ in Probability We first introduce the definition of -statistical convergence of order γ in probability for a sequence of rom variables as follows. Definition.. Let S,, P ) be a probability space X n } n N be a sequence of rom variables where each X n is defined on the same sample space S for each n) with respect to a given class of events a given probability function P : R. Then the sequence X n } n N is said to be -statistically convergent of order γ where 0 < γ ) in probability to a rom variable X where X : S R) if for any ε, δ > 0 β n α n + ) k [α n, β γ n ] : P X k X ε) δ} = 0, or equivalently β n α n + ) k [α n, β γ n ] : P X k X < ε) δ} = 0. In this case we write S γ P X n X ε) = 0 or S γ P X n X < ε) = P S γ or just X n X. The class of all sequences of rom variables which are statistically convergent of order γ in probability is denoted simply by P S γ. In Definition. if we take α n = β n = n, then X n } n N is said to be statistically convergent of order γ where 0 < γ ) in probability to a rom variable X. So -statistical convergence of order γ in probability is a generalization of statistical convergence of order γ in probability for a sequence of rom variables.

4 54 P. DAS, S. SOM, S. GHOSAL, AND V. KARAKAYA To show that this is indeed more general we will now give an example of a sequence of rom variables which is -statistically convergent of order in probability but is not statistically convergent of order in probability. Example.. Let a sequence of rom variables X n } n N be defined by, } with probability, if n = m for some m N, X n 0, } with probability P X n = 0) = n ) P X n = ) =, n if n m, for any m N. Let 0 < ε, δ <. Then we have P X n 0 ε) =, if n = m for some m N P X n 0 ε) = n, if n m for any m N. Let γ =, α n = n ) + ), β n = n for all n N. Then we have the inequality, ) k [n ) +, n d ] : P X n 0 ε) δ} = + 0, n n n P S as n where d is a finite positive integer. So X n 0. But n k n : P X n 0 ε) δ}. n n So the right h side does not tend to 0. This shows that X n } n N is not statistically convergent of order in probability to 0. S γ Theorem.. If a sequence of constants x n x then regarding a constant as a rom variable having one point distribution at that point, we may also write P S γ x. x n Proof. Let ε > 0 be any arbitrarily small positive real number. Then β n α n + ) k [α n, β γ n ] : x k x ε} = 0. Now let δ > 0. So the set K = k N : P x k x ε) δ} K where P S γ K = k N : x k x ε}. This shows that x n x. The following example shows that in general the converse of Theorem. is not true also shows that there is a sequence X n } n N of rom variables which is -statistically convergent in probability to a rom variable X but it is not -statistically convergent of order γ in probability for 0 < γ <.

5 A NOTION OF -STATISTICAL CONVERGENCE OF ORDER γ IN PROBABILITY 55 Example.. Let c be a rational number between γ γ. Let the probability density function of X n be given by, where 0 < x <, f n x) = 0, otherwise, if n = [m c ] for any m N, f n x) = nx n, n where 0 < x <, 0, otherwise, if n [m c ] for any m N. Now let 0 < ε δ <. Then P X n ε) =, if n = [m c ] for some m N, P X n ε) = ε ) n, if n [m c ] for any m N. Now let α n = β n = n. Consequently we have the inequality n c n γ n k [, γ n ] : P X k ε) δ} n c n k [, + γ n ] : P X k ε) δ} n γ + d ), n γ where d is a fixed finite positive integer. This shows that X n } n N is -statistically convergent of order γ in probability to but is not -statistically convergent of order γ in probability to whenever γ < γ this is not the usual -statistical convergence of order γ of real numbers. So the converse of Theorem. is not true. P S Also by taking γ =, we see that X n but Xn } n N is not -statistically convergent of order γ in probability to for 0 < γ <. Theorem. Elementary properties). We have the following. P S γ i) If X n X X n 0 < γ, γ. P S γ P S γ Y, then P X = Y } = for any γ, γ P S γ iv) Let g : R R be a continuous function 0 < γ γ. If X n where ii) If X n X Y n Y then cx n + dy n ) P Smaxγ,γ} cx + dy ) where c, d are constants 0 < γ, γ. iii) Let 0 < γ γ. Then P S γ P Sγ this inclusion is strict whenever γ < γ. P S γ X then gx n ) P Sγ gx). Proof.

6 56 P. DAS, S. SOM, S. GHOSAL, AND V. KARAKAYA i) Without loss of generality we assume γ γ. If possible let P X = Y } =. Then there exists two positive real numbers ε δ such that P X Y ε) = δ > 0. Then we have β n α n + β n α n + ) γ β n α n + ) γ k [α n, β n ] : P + β n α n + ) γ k [α n, β n ] : P X k X ε ) } X k Y ε ), } which is impossible because the left h it is not 0 whereas the right h it is 0. So P X = Y } =. ii) Proof is straightforward so is omitted. iii) The first part is obvious. The inclusion is proper as can be seen from Example.. iv) Proof is straightforward so is omitted. Remark.. In Theorem [] it was observed that m γ 0 m γ 0 this inclusion was shown to be strict for at least those γ, γ for which there is a k N such that γ < k < γ. But Example. shows that the inequality is strict whenever γ < γ. Theorem.3. Let 0 < γ, α, β) α, β ) are two pairs of sequences of positive real numbers such that [α n, β n] [α n, β n ] for all n N β n α n + ) γ εβ n α n + ) γ for some ε > 0. Then we have P S γ P Sγ α β. Proof. Proof is straightforward so is omitted. But if the condition of the Theorem.3 is violated then it may not be unique for two different α, β) s. We now give an example to show this. Example.3. Let α = n)!}, β = n + )!} α = n + )!}, β = n + )!}. Let us define a sequence of rom variables X n } n N by, } with probability P X k = ) =, P X k k = ) = ), k if n)! < k < n + )!,, } with probability P X k = ) = X k, P X k n = ) = ), k if n + )! < k < n + )!, 3, 3} with probability P X k = 3) = P X k = 3), if k = n)! k = n + )!. Let 0 < ε, δ < 0 < γ <. Then for the sequence α, β) P X k ε) =, if n)! < k < n + )!, k P X k ε) =, if n + )! < k < n + )!,

7 A NOTION OF -STATISTICAL CONVERGENCE OF ORDER γ IN PROBABILITY 57 Therefore, P X k ε) =, if k = n)! k = n + )!. n + )! n)! + ) k [n)!, n + )!] : P X γ k ε) δ} = 0. So X n P S γ. Similarly it can be shown that for the sequence α = n + )!}, β = n + )!}, X n P S γ α β. Definition.. Let S,, P ) be a probability space X n } n N be a sequence of rom variables where each X n is defined on the same sample space S for each n) with respect to a given class of events a given probability function P : R. A sequence of rom variables X n } n N is said to be -strong p-cesàro summable of order γ where 0 < γ p > 0 is any fixed positive real number) in probability to a rom variable X if for any ε > 0 P X β n α n + ) γ k X ε)} p = 0. k [α n,β n] P W γ,p In this case we write X n X. The class of all sequences of rom variables which are -strong p-cesàro summable of order γ in probability is denoted simply by P W γ,p. Theorem.4. i) Let 0 < γ γ, then P W γ,p P W γ,p. This inclusion is strict whenever γ < γ. ii) Let 0 < γ 0 < p < q <, then P W γ,q P W γ,p. Proof. i) The first part of this theorem is straightforward so is omitted. For the second part we will give an example to show that there is a sequence of rom variables X n } n N which is -strong p-cesàro summable of order γ in probability to a rom variable X but is not -strong p-cesàro summable of order γ in probability whenever γ < γ. Let c be a rational number between γ γ. We consider a sequence of rom variables:, } with probability, if n = [m c ] for some m N, X n 0, } with probability P X n = 0) = p n P X n = ) = p, n if n [m c ] for any m N. Then we have, for 0 < ε < P X n 0 ε) =, if n = [m c ] for some m N,

8 58 P. DAS, S. SOM, S. GHOSAL, AND V. KARAKAYA P X n 0 ε) = p n, if n [m c ] for any m N. Let α n = β n = n. So we have the inequality n γ k [,n ] n c n γ P X k 0 ε)} p n γ k [,n ] [ n c + P X k 0 ε)} p n γ + n γ )]. n 4 P W γ,p This shows that X n 0 but X n } n N is not -strong p-cesàro summable of order γ in probability to 0. ii) Proof is straightforward so is omitted. Theorem.5. Let 0 < γ γ. Then P W γ,p P S γ. Proof. Proof is straightforward so is omitted. So we can say that if a sequence of rom variables X n } n N is -strong p-cesàro summable of order γ in probability to X then it is -statistically convergent of order γ in probability to X i.e. P W γ,p P Sγ. But the converse of Theorem.5 is not generally true as can be seen from the following example. Example.4. Let a sequence of rom variables X n } n N be defined by,, } with probability, if n = mm for some m N, X n 0, } with probability P X n = 0) = p P X n n = ) = if n m m for any m N. Let 0 < ε < be given. Then P X n 0 ε) =, if n = m m for some m N p n, P X n 0 ε) = p n, if n mm for any m N.

9 A NOTION OF -STATISTICAL CONVERGENCE OF ORDER γ IN PROBABILITY 59 Let α n = β n = n P S γ. This implies X n 0 for each 0 < γ. Next let H = n N : n m m for any m N}. Then n γ k [,n ] = n γ = n γ > n γ k [,n ] k H k [,n ] k H n k= > n γ, k P X k 0 ε)} p P X k 0 ε)} p + k + n γ k [,n ] k / H n γ k [,n ] k / H P X k 0 ε)} p since n k= k > n, for n. So X n is not -strong p-cesàro summable of order γ in probability to 0 for 0 < γ. Theorem.6. i) For γ =, P W,p = P S. ii) Let g : R R be a continuous function 0 < γ γ. If X n then gx n ) P W γ gx). P W γ X, Proof. For i) ii) the proof is straightforward so is omitted. Theorem.7. Let α n } n N, β n } n N be two non-decreasing sequences of positive numbers such that α n β n α n+ β n+ 0 < γ γ. Then P S γ P S γ iff inf βn α n ) >. Such a pair of sequence exists: take α n = n! β n = n + )!.) Proof. First of all suppose that inf βn P S α n ) > let X γ n X. As inf βn α n ) >, for each δ > 0 we can find sufficiently large r such that β r α r + δ) therefore ) γ βr α r β r ) γ δ. + δ

10 60 P. DAS, S. SOM, S. GHOSAL, AND V. KARAKAYA Now for each ε, δ > 0 we have [β n ] k [β n] : P X γ k X ε) δ} = [β n ] k β γ n : P X k X ε) δ} k β n : P X k X ε) δ} β γ n δ + δ ) γ β n α n + ) γ k [α n, β n ] : P X k X ε) δ}. Hence the result follows. Now if possible, suppose that inf β n α n ) =. So for each j N we can choose a subsequence such that β rj) α rj) < + β rj) j β rj ) j. Let I rj) = [α rj), β rj) ]. We define a sequence of rom variables by, } with probability, if n I rj) where j N, X n 0, } with probability P X n = 0) = P X n n = ) =, n if n / I rj) for any j N. Let 0 < ε, δ <. Now P X n 0 ε) =, if n I rj) where j N, P X n 0 ε) = n, if n / I rj) for any j N. Now β rj) α rj) k [α +) γ rj), β rj) ] : P X n 0 ε) δ} = β rj) α rj) +) β rj) α rj) +) γ as j. But as β rj) α rj) k [α +) γ rj), β rj) ] : P X n 0 ε) δ} is a subsequence of the sequence β r α r+) k [α γ r, β r ] : P X n 0 ε) δ}, this shows that X n is not -statistically convergent of order γwhere 0 < γ ) in probability to 0. Finally let γ =. If we take t sufficiently large such that α rj) < t β rj) then we observe that t t P X k 0 ε) β rj ) + β rj) α rj) + + α rj) t + + t } β rj ) + β rj) α rj) + + β rj) α rj) t + + t } j + + t + + t } 0, if j, t. k= P S This shows that X n 0. But this is a contradiction as P S γ P S γ where 0 < γ γ ). We conclude that inf βn α n ) must be >.

11 A NOTION OF -STATISTICAL CONVERGENCE OF ORDER γ IN PROBABILITY 6 Theorem.8. Let α = α n } n N, β = β n } n N be two non-decreasing sequences P S of positive numbers. Let X γ P S γ n X X n Y for 0 < γ γ. If inf βn α n ) > then P X = Y } =. Proof. Let ε > 0 be any small positive real number if possible let P X Y ε) = δ > 0. Now we have the inequality P X Y ε) P X n X ε )} + P X n Y ε )} k [α n, β n ] : P X Y ε) δ} k [α n, β n ] : P X k X ε ) δ } k [α n, β n ] : P X k Y ε ) δ } k [α n, β n ]:P X Y ε) δ} k [α n, β n ] : P X k X ε ) } + k [α n, β n ] : P X k Y ε ) } k [α n, β n ]:P X Y ε) δ} k [β n ] : P X k X ε ) } + k [α n, β n ] : P X k Y ε ) } β n α n ) k [β n ] : P X k X ε ) } + k [α n, β n ] : P X k Y ε ) } ) γ βn α n β n [β n ] γ k [β n ] : P X k X ε ) } + β γ k [α n, β n ] : P X k Y ε ) n } ) γ βn α n β n [β n ] γ k [β n ] : P X k X ε ) δ ) γ βn α n + + } β n β n α n + ) γ k [α n, β n ] : P X k Y ε ) } α ) γ n β n [β n ] γ k [β n ] : P X k X ε ) δ + } α n + ) γ β n β n β n α n + ) γ k [α n, β n ] : P X k Y ε ). }

12 6 P. DAS, S. SOM, S. GHOSAL, AND V. KARAKAYA Taking n ) on both sides we see that the left h side does not tend to zero β since inf n α n > but the right h side tends to zero. This is a contradiction. So we must have P X = Y } =. 3. -Statistical Convergence of Order γ in r th Expectation Definition 3.. Let S,, P ) be a probability space X n } n N be a sequence of rom variables where each X n is defined on the same sample space S for each n) with respect to a given class of events a given probability function P : R. Then the sequence X n } n N is said to be -statistically convergent of order γ where 0 < γ ) in r th expectation to a rom variable X where X : S R) if for any ε > 0 β n α n + ) k [α n, β γ n ] : E X k X r ) ε} = 0, provided E X n r ) E X r ) exists for all n N. In this case we write S γ ES γ,r E X n X r ) = 0 or by X n X. The class of all sequences of rom variables which are -statistically convergent of order γ in r th expectation is denoted simply by ES γ,r. ES γ,r P S γ Theorem 3.. Let X n X for any r > 0 0 < γ ). Then X n X, i.e. -statistical convergence of order γ in r th expectation implies -statistical convergence of order γ in probability. Proof. The proof can be easily obtained by using Bienayme-Tchebycheff s inequality. The following example shows that in general the converse of Theorem 3. is not true. Example 3.. We consider the sequence of rom variables X n } n N defined by 0, } with probability P X n = 0) = P X n = ), if n = m for some m N, X n 0, n} with probability P X n = 0) = ) P X n r n = ) =, n r if n m for any m N, where r > 0. Now let 0 < ε, δ <. Then we have P X n 0 ε) =, if n = m for some m N, P X n 0 ε) = n r, if n m for any m N.

13 A NOTION OF -STATISTICAL CONVERGENCE OF ORDER γ IN PROBABILITY 63 Let γ =, α n = n ) + ), β n = n. Then we have the inequality ) k [n ) +, n d ] : P X n 0 ε) δ} = + 0 n n n as n, where d is a finite positive integer. So X n But E X n 0 r ) = P S 0. if n = m for some m N,, if n m for any m N. This shows that S - E X n 0 r ) 0, i.e. X n is not -statistically convergent of order in rth expectation to 0. Theorem 3.. Let X n } n N be a sequence of rom variables such that P S γ P X n M) = for all n some constant M > 0. Suppose that X n X. ES γ,r Then X n X for any r > 0. Theorem 3.3. ES γ,r ES γ,r i) Let X n X X n Y for all r > 0 0 < γ ). P X = Y ) = provided X X n ) 0 Y n Y ) 0. ES γ,r ES γ,r ii) Let X n X Y n Y for all r > 0 0 < γ ). X n + Y n ) ESγ X + Y ) provided X X n ) 0 Y n Y ) 0. Then Then 4. -Statistical Convergence of Order γ in Distribution Definition 4.. Let S,, P ) be a probability space X n } n N be a sequence of rom variables where each X n is defined on the same sample space S for each n) with respect to a given class of events a given probability function P : R. Let F n x) be the distribution function of X n for all n N. If there exist a rom variable X whose distribution function is F x) such that the sequence F n x)} n N is -statistically convergent of order γ to F x) at every point of continuity x of F x) then X n } n N is said to be -statistically convergent of order γ in distribution to X DS γ we write X n X. Theorem 4.. Let X n } n N be a sequence of rom variables. Also let f n x) = P X n = x) be the probability mass function of X n for all n N fx) = P X = x) be the probability mass function of X. If f n x) Sγ DS γ fx) for all x then X n X. Proof. Proof is straightforward, so omitted.

14 64 P. DAS, S. SOM, S. GHOSAL, AND V. KARAKAYA Proposition 4.. Let a n } n N, b n } n N be two sequences of real numbers such that a n b n for all n N. Then S γ a n S γ b n S γ a n S γ b n. Here S γ Sγ denotes -statistical it inferior of order γ -statistical it superior of order γ of the respective real sequences here we use the same definition as in [3] but here natural density is replaced by the density of order γ. Proof. Proof is straightforward, so omitted. P S γ Theorem 4.. Let X n } n N be a sequence of rom variables. If X n X then DS γ X n X. That is, -statistical convergence of order γ in probability implies -statistical convergence of order γ in distribution. Proof. Let F n x) F x) be the probability distribution functions of X n X respectively. Let x < y. Now X x) X n y, X x) + X n > y, X x) X x) X n y) + X n > y, X x) P X x) P X n y) + P X n > y, X x) P X x) P X n y) + P X n X > y x) F x) F n y) + P X n X > y x) S γ F x) Sγ F ny) F x) S γ F ny), similarly following same kind of steps by taking y < z we get S γ F ny) F z). Let y be a point of continuity of the function F x). Then F x) = F z) = F y). x y z y+ Hence we get S γ F ny) = F y). Hence the result follows. Now we will show that the converse of the Theorem 4. is not necessarily true i.e. -statistical convergence of order γ in distribution may not implies -statistical convergence of order γ in probability. For this we will construct an example as follows. Example 4.. Let α n = n ) +) β n = n consider rom variables X, X n where n is the first [h r c ] integers in the interval I r = [α r, β r ] where h r = β r α r + ) 0 < c < ) having identical distribution. Let γ be a real number in 0, ] such that γ > c. Let the spectrum of the two dimensional rom variables X n, X) be, 0),, ),, 0),, ) with probability P X n =, X = ) = 0 = P X n =, X = 0)

15 A NOTION OF -STATISTICAL CONVERGENCE OF ORDER γ IN PROBABILITY 65 P X n =, X = ) = = P X n =, X = 0). Hence, the marginal distribution of X n is given by X n = ii =, ) with p.m.f. f Xn ) = f Xn ) = the marginal distribution of X is given by X = ii = 0, ) with p.m.f. f X 0) = f X ) =. Next, we consider rom variables X, X n where n is other than the first [h c r ] integers in the interval I r = [α r, β r ]) having identical distribution the spectrum of the two dimensional rom variables X n, X) be 0, 0), 0, ),, 0),, ) with probability P X n = 0, X = 0) = 0 = P X n =, X = ) P X n =, X = 0) = = P X n = 0, X = ). Hence, the marginal distribution of X n is given by X n = i i = 0, ) with p.m.f. f Xn 0) = f Xn ) = the marginal distribution of X is given by X = i i = 0, ) with p.m.f. f X 0) = f X ) =. Let n is the first [h c r ] integers in the interval I r F n x) be the probability distribution function of X n then 0, if x <, F n x) =, if x <,, if x. Next let, n is other than the first [h c r ] integers in the interval I r F n x) F x) be the probability distribution function of X n X respectively, then 0, if x < 0, F n x) = F x) =, if 0 x <,, if x. We consider the interval [, 0). It is sufficient to prove that the sequence y n } n N define below is -statistically convergent of order γ to 0. Now we define a sequence y n } n N by y n =, if n is the first [h r c ] integers in the interval I r, 0, if n is other than the first [h c r ] integers in the interval I r. It is quite clearly that y n } n N is -statistically convergent of order γ to 0, this implies F n x) Sγ F x) for all x R but F n x) F x) for all x [, 0) in DS γ ordinary sense). This shows that X n X. For any 0 < ɛ <, we have P X n X ε) =, for all, n I r, r Z +. But this shows that S γ P X n X ɛ) 0. This shows that the sequence X n } n N is not -statistically convergent of order γ in probability to X.

16 66 P. DAS, S. SOM, S. GHOSAL, AND V. KARAKAYA The statement of Theorem 4. does not depend on the it infimum of the sequence q r = βr α r of the sequence α n β n. Even if inf q r =, Theorem 4. will hold. Theorem 4.3. Let α n } n N β n } n N be two increasing sequences of positive real numbers such that α n β n α n+ β n+, β n α n ) as n 0 < γ γ. If inf βn P S α n ) > then X γ DS γ n X implies X n X. Proof. The proof is straightforward, so omitted. Acknowledgements. The research of the second author is funded by UGC Research, HRDG, India. We would like to thank the respected referees for their careful reading of the paper for giving several valuable suggestions which have improved the presentation of the paper. References [] H. Aktuglu, Korovkin type approximation theorems proved via -statistical convergence, J. Comput. Appl. Math ), [] S. Bhunia, P. Das S. Pal, Restricting statistical convergence, Acta Math. Hungar. 34 ) 0), [3] R. Çolak, Statistical convergence of order α, Modern Methods in Analysis its Applications, Anamaya Pub 00), New Delhi, India, 9. [4] J. Connor, The statistical strong p-cesaro convergence of sequences, Analysis 8 ) 988), [5] J. Connor, J. Fridy J. Kline, Statistically pre-cauchy sequences, Analysis 4 994), [6] P. Das, S. Ghosal E. Savas, On generalization of certain summability methods using ideals, Appl. Math. Lett. 49) 0), [7] P. Das, S. Ghosal, S. Som, Statistical convergence of order α in probability, Arab J. Math. Sci. ) 05), [8] P. Das, S. Ghosal S. Som, Different types of quasi weighted -statistical convergence in probability, Filomat, 35) 07), [9] H. Fast, Sur la convergence statistique, Colloquium Math. 95), [0] R. Freedman, M. Raphael J. Sember, Some Cesaro-type summability spaces, Proc. Lond. Math. Soc. 373) 978), [] J. Fridy, On statistical convergence, Analysis 5 985), [] J. Fridy C. Orhan, Lacunary statistical convergence, Pacific. J. Math ), [3] J. Fridy C. Orhan, Statistical it superior it inferior, Proc. Amer. Math. Soc. 5) 997), [4] S. Ghosal, Statistical convergence of a sequence of rom variables it theorems, Appl. Math. 584) 03), [5] S. Ghosal, Generalized weighted rom convergence in probability, Appl. Math. Comput ), [6] T. Šalát, On statistically convergent sequences of real numbers, Math. Slovaca ), [7] J. Schoenberg, The integrability of certain functions related summability methods, Amer. Math. Monthly ), [8] H. Sengul M. Et, On lacunary statistical convergence of order α, Acta Math. Sci. Ser. B Engl. Ed. 34) 04),

17 A NOTION OF -STATISTICAL CONVERGENCE OF ORDER γ IN PROBABILITY 67 [9] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math. 95), [0] A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, 979. Department of Mathematics Jadavpur University Kolkata West Bengal, India address: P. Das) address: S. Som) School of Sciences Netaji Subhas Open University Kalyani Nadia-7435 West Bengal, India address: S. Ghosal) 3 Department of Mathematical Engineering Yildiz Technical University Davutpasa Campus Esenler Istanbul, Turkey address: vkkaya@yildiz.edu.tr V. Karakaya)