Non linear functionals preserving normal distribution and their asymptotic normality

Transcription

1 Armenian Journal of Mathematics Volume 10, Number 5, 018, 1 0 Non linear functionals preserving normal distribution and their asymptotic normality L A Khachatryan and B S Nahapetian Abstract We introduce sufficiently wide classes of non linear functionals preserving normal Gaussian) distribution and establish various conditions under which a sequence of such functionals is asymptotically normal As a consequence, we obtain a generalization and sharpening of nown results on the central limit theorem for weighted sums linear functionals) of independent random variables Key Words: normal distribution, central limit theorem, non linear functional on several variables Mathematics Subect Classification 010: 60F05, 6007, 6Bxx, 46N30 Introduction Limit theorems for sums of independent random variables form a complete theory which presents one of the main parts of the probability theory see, for instance, [4, 11]) At the same time, the question of the validity of the central limit theorem CLT) for wider classes of random processes is still very actual There are two main directions for researches: the validity of the CLT for sums of dependent random variables, for example, processes with wealy dependent components [, 6, 10]), martingales [5]), and the asymptotic normality of different classes of non linear functionals on independent as well as dependent random variables The present paper is concerned with the second problem Namely, we introduce sufficiently wide classes of non linear functionals on random variables which have the following property: in the case when random variables are independent and standard normally Gaussian) distributed, the distribution of the functional is standard Gaussian too, or, in our terminology, the functional preserves the normal distribution Section 1) In the second section, we consider functionals, which preserve the normal distribution, on independent, but not necessarily Gaussian random variables, and obtain conditions under which a sequence of such functionals is asymptotically normal 1

2 L A KHACHATRYAN AND B S NAHAPTIAN As a consequence, we also obtain a generalization and sharpening of nown results on the CLT for weighted sums linear functionals) of independent random variables Section 3) 1 Functionals preserving the normal distribution Denote by f n f n x 1, x,, x n ) a real valued functional defined in R n, n We say that the functional f n preserves the normal distribution, if for random variables ξ 1, ξ,, ξ n which are independent and standard normally distributed, the random variable f n ξ 1, ξ,, ξ n ) has the standard normal distribution N 0, 1) too Here are several simple examples of functionals preserving the normal distribution see, for instance, [8] and [1]) xample 1 f x 1, x ) x 1 + x xample f x 1, x ) x1 x x 1 + x xample 3 f 3 x 1, x, x 3 ) x 1x + x x 1 It is obvious that any superposition of functionals preserving the normal distribution is again a functional which preserves the normal distribution This observation gives us an approach to the construction sequences f n, n, of such functionals For instance, xample 1 gives rise to the following sequence of functionals preserving the normal distribution f n x 1, x,, x n ) f x 1, f n 1 x,, x n )) x 1 + x ) + x 3 ) + + x n 3 ) + x n 1 + x n n, n > ) n 1 If we consider the functional ˆf x 1, x ) x 1x x 1 + x similar to the one in the xample, we obtain a sequence ˆf n x 1, x,, x n ) ˆf x 1, ˆf n 1 x,, x n )) x x i 1in : i, n >,

3 NON LINAR FUNCTIONALS PRSRVING NORMAL DISTRIBUTION AND THIR ASYMPTOTIC NORMALITY 3 of random variables normally distributed with parameters 0 and n 1 Hence, functionals f n x 1, x,, x n ) n x x i 1in : i, n >, preserve the normal distribution Starting from the functional from the third example, we obtain the following sequence of functionals preserving the normal distribution f n x 1, x,, x n ) f 3 x 1, x, f n 1 x 1, x 3, x 4,, x n )) x 1x + x 1x 3 x 1 x x x x 1) + + x 1 x n 1 3/ 1 + x 1) + x n n )/ 1 + x 1), n )/ n > 3 Sequences of preserving the normal distribution functionals can be obtained in other ways For instance, xample 1 is a particular case of linear functionals weighted sums) with coefficients, 1, n, ie f n x 1, x,, x n ) x The classical case is obtained for n 1/, 1, n For linear functionals the following statement is true Proposition 1 The linear functional f n with coefficients, 1, n, preserves the normal distribution if and only if ) 1 1) Proof Let ξ 1, ξ,, ξ n be independent random variables with the standard normal distributions Consider the characteristic function ϕ n of the random variable ξ ϕ n t) exp it ξ } e itαn) ξ

4 4 L A KHACHATRYAN AND B S NAHAPTIAN To prove the proposition we need to show that ϕ n t) e t /, t R Since ξ N 0, ), we have Hence ϕ n t) } exp it ξ exp 1 e αn) t) / exp t From here it follows that ϕ n t) e t / if and only if ) } t ) } ) 1 Next proposition shows how one can construct linear functionals preserving the normal distribution using a superposition Proposition Consider the functional and the superposition form f x 1, x ) αx 1 + βx, α, β R\0, 1}, f n x 1, x,, x n ) f x 1, f n 1 x, x 3,, x n )), n > ) 1 For any n, f n is the linear functional with coefficients of the n β n 1, αβ 1, 1 < n 1 3) Conversely, any linear functional f n with coefficients given by 3) can be represented as the superposition ) where f has coefficients α and β The linear functional ) preserves the normal distribution if and only if the coefficients α and β of the functional f are such that α + β 1 Proof It is easy to see that f x 1, f n 1 x, x 3,, x n )) x, where, 1 n, are some coefficients Let us show that these coefficients have the form 3) Indeed, for n 3 we have f 3 x 1, x, x 3 ) f x 1, f x, x 3 )) αx 1 + αβx + β x 3

5 NON LINAR FUNCTIONALS PRSRVING NORMAL DISTRIBUTION AND THIR ASYMPTOTIC NORMALITY 5 Assume now that the statement is true for the functional f n 1 Then n 1 f n x 1, x,, x n ) f x 1, f n 1 x, x 3,, x n )) αx 1 + β α n 1) x +1 αx 1 + β n ) n 1 αβ 1 x +1 + β n x n αβ 1 x + β n 1 x n, and hence coefficients of f n have the form 3) Conversely, for any linear functional f n with coefficients given by 3) we can write f n x 1, x,, x n ) αx 1 + β n n 1 x αβ 1 x + β n 1 x n ) αβ 1 x +1 + β n x n f x 1, f n 1 x, x 3,, x n )) To prove the second statement of the proposition note that for any n we have ) n 1 α β 1) ) + β n 1 α 1 β ) n 1 + β ) n 1 1 β If α + β 1 then the right hand side of the expression above equals to 1, and hence for f n condition 1) holds On the other hand, if linear functionals f n with coefficients 3) preserve the normal distribution then condition 1) is fulfilled for n as well, and hence α + β 1 Functional f 3 from the xample 3 can be generalized in the following way xample 4 f n+1 x 0, x 1, x,, x n ) x 0x 1 + x + + x n n 1 + x 0, n We can write f n+1 x 0, x 1, x,, x n ) x 0x 1 + x + + x n n 1 + x 0 x 0 x 1 + n n 1 + x 0 1 n 1 + x 0 x Note that for any x 0 R ) ) x 0 1 x 0 + n 1 + x 0 n 1 + x 0 n 1 + x 0 + n 1 n 1 + x 0 1

6 6 L A KHACHATRYAN AND B S NAHAPTIAN Below we prove a general result, from which, particularly, one can see that considered functionals preserve the normal distribution The main class of non linear functionals considered in the paper is defined as follows Let x), 1, n, x R, be a set of non zero functions, and put f n+1 x 0, x 1, x,, x n ) f x 0 n x 1, x,, x n ) x 0 )x 4) In the next theorem, a condition for such functional to preserve the normal distribution is introduced Theorem 1 The functional 4) preserves the normal distribution if z)) 1 for any z R 5) Proof Let random variables ξ 0, ξ 1, ξ,, ξ n be independent and standard normally distributed We will show that ) 1 ) ξ 0 )ξ 0, ξ 0 )ξ )!, 1,,! Since the normal distribution is uniquely determined by its moments, from here it will follow that ξ 0 )ξ N 0, 1) First consider odd moments Taing into account the independence of ξ 0, ξ 1, ξ,, ξ n, we can write ) 1 ξ 0 )ξ 0m 1,m,,m n 1: m 1 +m ++m n 1 0m 1,,m n 1: m 1 ++m n 1 1)! m 1!m! m n! 1)! m 1!m! m n! ξ m i i ) mi i ξ 0 ) ξ m i i n ) ) mi i ξ 0 ) It remains to note that since m 1 + m + + m n 1, for any numbers 0 m 1,, m n 1 there exists i, 1 i n, such that m i is odd, and hence ξ m i i 0

7 NON LINAR FUNCTIONALS PRSRVING NORMAL DISTRIBUTION AND THIR ASYMPTOTIC NORMALITY 7 Now consider even moments We have ) ξ 0 )ξ 0m 1,m,,m n: m 1 +m ++m n )! m 1!m! m n! ξ m i i n ) ) mi i ξ 0 ) If there is at least one odd number among m 1, m,, m n, then n 0m 1,,m n: m 1 ++m n ξ m i i 0 Hence only those summands remain in the sum for which all numbers m 1, m,, m n are even Therefore we can write ) ξ 0 )ξ )! n ) ) ξ m i i mi i ξ 0 ) m 1 )! m n )! 0m 1,,m n: m 1 ++m n )! n ) m i )! m 1 )! m n )! m i mi! ) mi i ξ 0 ) )!! )!! 0m 1,,m n: m 1 ++m n i ξ 0 )) )! m 1! m n! It remains to note that due to the condition 5), for any ) ) mi i ξ 0 ) i ξ 0 )) ) 1 It is not difficult to see that in the proof of the theorem above we do not use any condition on the distribution of ξ 0 Hence the result of the previous theorem can be generalized in the following form Theorem Let ζ, ξ 1, ξ,, ξ n be independent random variables, and let ξ N 0, 1), 1, n The random variable f ζ nξ 1, ξ,, ξ n ) i ζ)ξ i

8 8 L A KHACHATRYAN AND B S NAHAPTIAN has the standard normal distribution if condition 5) is fulfilled xample 4 permits the following generalization xample 5 Let g : R R be a Borel function, and put n z) gz) n 1 + g z), αn) z) 1 n 1 + g z), 1 n 1 Then functionals f z nx 1, x,, x n ) n 1 preserve the normal distribution Indeed, for any fixed n x n 1 + g z) + ) n 1 z) n 1 + g z) + x n gz) n 1 + g z), n, g z) n 1 + g z) 1 Hence the condition 5) is fulfilled for any z R Further generalization of this example one can find in [9] Asymptotic normality of sequences of functionals preserving the normal distribution In this section we consider functionals f n on independent random variables η 1, η,, η n which are not necessarily Gaussian We say that a sequence of functionals f n η 1, η,, η n ), n, is asymptotically normal if their distributions converge to the standard normal one as n Conditions under which a sequence of preserving the normal distribution functionals 4) is asymptotically normal are given in the following technical lemma Lemma 1 Let functions, 1, n, satisfying condition 5), be such that for independent random variables ζ, ξ 1,, ξ n, where ξ N 0, 1), and any ε > 0, ) ) ) ζ)ξ I ζ)ξ > ε 0 as n

9 NON LINAR FUNCTIONALS PRSRVING NORMAL DISTRIBUTION AND THIR ASYMPTOTIC NORMALITY 9 Then for any independent random variables η 1,, η n which are independent of ζ and such that η 0, Dη 1, 1, n, and for any ε > 0 ) ) ) ζ)η I ζ)η > ε 0 as n, the sequence of functionals f ζ nη 1, η,, η n ) ζ)η, n, is asymptotically normal Proof We need to show that } exp it ζ)η e t / 0 as n Let the standard normally distributed random variables ξ 1,, ξ n be independent of η 1,, η n Since the functionals under consideration fn ζ preserve the normal distribution we have exp Hence we can write exp it n Denote it exp it n ζ)ξ } e t /, n ζ)η } and for any, 1 < < n, e t / ζ)η } exp ζ n) 1 f ζ n0, ξ,, ξ n ) it n ζ)ξ, n 1 ζ n n) fnη ζ 1,, η n 1, 0) ζ)η, 1 ζ n) fnη ζ 1,, η 1, 0, ξ +1,, ξ n ) ζ)η + ζ)ξ } +1 ζ)ξ

10 10 L A KHACHATRYAN AND B S NAHAPTIAN Then exp it 1 ζ)η } exp it ζ)ξ } } )}) exp itζ n) + ζ)η ) exp itζ n) + ζ)ξ 1 e itζn) e itαn) ζ)η e itζn) e itαn) ζ)ξ Since for any e itζn) ζ)η e itζn) ζ) η 0 e itζn) ζ) ξ e itζn) ζ)ξ, e itζn) ζ)η ) e itζ n) ζ) ) η and e itζn) 1 e itζn) 1, further we can write e itζn) eitζn) 1 e itζn) eitαn) eitαn) e itαn) ζ)η e itζn) e itαn) e itαn) ) ζ) ξ e itζn) e itαn) ζ)η 1 it ζ)ξ ζ)ξ n) 1 itα ζ)ξ it) ζ)η 1 it ζ)ξ 1 it ζ)ξ it) ζ)ξ ), ζ)η it) αn) ζ)η ) ) αn) ζ)ξ ) ) ζ)η it) αn) ζ)η ) + αn) ζ)ξ )

11 NON LINAR FUNCTIONALS PRSRVING NORMAL DISTRIBUTION AND THIR ASYMPTOTIC NORMALITY 11 Using the well-nown estimation } N it) eit! min t N t N+1,, N 0, 1,, 6) N! N + 1)! 0 for any ε > 0 we obtain 1 t 3 3! eitαn) ζ)η 1 it ζ)η it) : ζ)η ε t 3 6 ε 1 t 3 6 ε + t Similarly, ζ)η 3 + t! ζ) ) η + t 1 1 ζ)η eitαn) t 3 6 ε + t 1 ) I αn) : ζ)η >ε ζ)η ) ζ)η ζ)η > ε) ) ζ)ξ 1 it ζ)ξ it) 1 ζ)ξ ) I αn) ) I ζ)η ζ)ξ ) ζ)ξ > ε) ) ζ)η > ε) ) Hence exp it n ζ)η } e t / t 3 3 ε+ +t n 1 +t n 1 ζ)η ζ)ξ which completes the proof ) I ) I ζ)η > ε) ) + ζ)ξ > ε) ), An application of the Lemma 1 brings to the following result

12 1 L A KHACHATRYAN AND B S NAHAPTIAN Theorem 3 Let functions, 1, n, satisfying condition 5), be such that max z) 0 as n sup 1n z R Then for any independent random variables ζ, η 1,, η n such that η 0, Dη 1, 1, n, and for some δ > 0 the sequence of functionals sup η +δ C δ <, 1n f ζ nη 1, η,, η n ) is asymptotically normal ζ)η, n, Proof We need to chec that conditions of the Lemma 1 are fulfilled For any ε > 0 we have +δ α n) +δ ) ) ζ)η ζ)η I ζ)η > ε > > ε δ ) ) ) ζ)η I ζ)η > ε, and hence ) ) ) ζ)η I ζ)η > ε 1 ε δ 1 n Then ) ) ) ζ)η I ζ)η > ε C δ ε δ C δ ε δ max sup 1n z R z) δ ζ) +δ η +δ C δ α ε n) δ ζ) +δ, ) C δ ζ) max ε δ ζ) +δ sup 1n z R δ z) ) 0 as n Similarly, for independent standard normally distributed random variables ξ 1, ξ,, ξ n, which are independent of ζ, we have ) ) ) ζ)ξ I ζ)ξ > ε 3+δ)/4 ζ) +δ, ε δ

13 NON LINAR FUNCTIONALS PRSRVING NORMAL DISTRIBUTION AND THIR ASYMPTOTIC NORMALITY 13 and hence as n 3+δ)/4 ε δ ) ) ) ζ)ξ I ζ)ξ > ε max sup 1n z R δ z) ) 0 Another type of conditions on the functions z), z R, 1, n, under which the sequence of functionals given by 4) is asymptotically normal, is introduced in the following two theorems Theorem 4 Let random variable ζ and functions, 1, n, be such that ζ) 1 0 as n 7) n αn) Let η 1, η,, η n be independent random variables with finite second moments which are independent of ζ too and for which the CLT holds Then the sequence of functionals fnη ζ 1, η,, η n ) ζ)η, n, is asymptotically normal Proof For any n we can write fnη ζ 1, η,, η n ) ζ)η 1 n η + ζ) 1 n ) η Since for random variables η 1, η,, η n the CLT is valid, the first summand on the right hand side of the obtained expression is asymptotically normal Therefore to complete the proof we need to show that the second summand tends to 0 in probability Note that η ) η 1/ C < for any 1, n Hence due to the Chebychev inequality for any ε > 0 we have P 1 ε as n ζ) 1 n ) η > ε αn) ) ζ) 1 η i C n ε 1 ε αn) ζ) 1 n ) η ζ) 1 0 n

14 14 L A KHACHATRYAN AND B S NAHAPTIAN xample 5 continuation) Let us show that functions, 1, n, considered in the xample 5 satisfy the condition 7) Indeed, let function g be such that g ζ) < For any fixed n we have ζ) 1 n αn) gζ) n 1 + g ζ) n 1) n n 1 + g ζ) 1 n For the first summand in the right hand side of this expression we obtain gζ) n 1 + g ζ) 1 gζ) as n, n n 1 n since gζ) g ζ)) 1/ < For the second summand we can write 1 n 1) n 1 + g ζ) 1 n n 1 + g ζ) n 1) n n n 1 + g ζ) n 1) n n 1 + g ζ)) n ) n + n 1 + g ζ) n 1 + g ζ) n 1 1 g ζ) 0 n n 1 as n Theorem 5 Let functions, 1, n, satisfy the condition 5) and let random variable ζ be such that α max n) ζ) 0 as n 0 1n and for some δ, 0 < δ < 1/, max n δ) ζ)) C < Let η 1, η,, η n be independent random variables which are independent of ζ too and such that η 0, Dη 1, 1, n Then the sequence of functionals f ζ nη 1, η,, η n ) is asymptotically normal ζ)η, n,

15 NON LINAR FUNCTIONALS PRSRVING NORMAL DISTRIBUTION AND THIR ASYMPTOTIC NORMALITY 15 Proof Let ξ 1, ξ,, ξ n be independent standard normally distributed random variables which are also independent of ζ, η 1, η,, η n We have f ζ nη 1, η,, η n ) ζ)η ζ)ξ + ζ)η ξ ) Since the condition 5) is fulfilled, due to the Theorem, ζ)ξ N 0, 1) Hence to prove the theorem we need to show that the last summand in the expression above converges in probability to 0 as n Successively applying Chebyshev and Minowsi inequalities, for any ε > 0 we can write ) 1+δ P ζ)η ξ ) > ε P ζ)η ξ ) > ε 1+δ 1 ε 1+δ ζ)η ξ ) 1+δ ) 1 1+δ 1 ε 1+δ 1+δ ζ)η ξ ) 1+δ Since random variables ζ, η and ξ are independent, for any 1, n we have ζ)η ξ ) 1+δ ζ) 1+δ η ξ 1+δ Here η ξ 1+δ 1+δ η ξ ) ) 1+δ, by virtue of Hölder s inequality ζ) 1+δ α n) ζ) ζ) δ) 1 ) 1 ζ) ζ) δ, and ζ) δ δ ζ) since 0 < δ < 1 Hence 1 1+δ ζ)η ξ ) 1+δ 1 1+δ) ) δ ζ) 1+δ ζ) α max n) ζ) 1n δ 1+δ 1 1+δ) ) α ζ) C max n) ζ) 1n ) δ 1+δ

16 16 L A KHACHATRYAN AND B S NAHAPTIAN Finally we obtain ) P ζ)η ξ ) > ε C ) 1+δ δ α max ε n) ζ) ) 0 1n as n 3 Remars on asymptotic normality of sequences of linear functionals In this section we consider linear functionals f n η 1, η,, η n ) η, n Conditions under which a sequence of such functionals weighted sums) is asymptotically normal were obtained in several wors For example, Weber in [13] introduced conditions on the 4-th power of the coefficients as well as existence of η p for p > 4 Fisher [3]) and Kevei [7]) used the appropriate rate of convergence to 0 for the coefficients It is also worth noting that in the mentioned papers only identically distributed random variables η 1, η,, η n were considered Below we formulate the CLT for linear functionals on independent but not necessarily identically distributed random variables under the condition of the finiteness of their +δ)- th moments for some δ > 0 and the condition on squares of corresponding coefficients Theorem 6 Let coefficients, 1, n, of linear functionals f n satisfy the condition 1) and be such that max 0 as n 1n Let η 1, η,, η n be independent random variables such that η 0, η 1, 1, n, and for some δ > 0 sup η +δ C δ < 1n Then the sequence of linear functionals f n η 1, η,, η n ), n, is asymptotically normal This theorem can be obtained as a direct corollary of the Theorem 3 for z), 1 n On the other hand, it can be easily proved by the use of the following general lemma

17 NON LINAR FUNCTIONALS PRSRVING NORMAL DISTRIBUTION AND THIR ASYMPTOTIC NORMALITY 17 Lemma Let the coefficients, 1, n, of the linear functionals f n satisfy the condition 1) and be such that for independent random variables η 1, η,, η n with η 0, Dη 1, 1, n, and any ε > 0 ) ) ) η I η > ε 0 as n Then the sequence of linear functionals f n η 1, η,, η n ), n, is asymptotically normal The proof of the Lemma can be obtained as a corollary of the CLT for a series scheme of random variables independent in each series see, for instance, Theorem 7 in [1]) An application of the method used in the proof of Lemma 1 allows us to give quite simple proof of this result Proof of the Lemma First let us note that using mathematical induction, it is not difficult to show that for any two finite sequences of complex numbers a 1, a,, a N and b 1, b,, b N such that a 1, b 1, 1 N, the following estimation is valid N a 1 N b 1 N a b 8) 1 Let ξ 1, ξ,, ξ n be independent standard normally distributed random variables which are independent of η 1, η,, η n as well Then ξ N 0, 1), and hence with the use of 8) we can write eit η e t / e itαn) η e itαn) ξ e itαn) η e itαn) ξ Since for any η ξ 0, η ) ξ ) ), ξ ) 3 0,

18 18 L A KHACHATRYAN AND B S NAHAPTIAN further we have + e itαn) eitαn) η e itαn) ξ ξ 1 it ξ it)! eitαn) η 1 it ξ ) it)3 ξ ) 3 3! η it) η ) +! With application of the estimation 6) for any ε > 0 we can write t 3 3! eitαn) : η ε t 3 6 ε + t Since ξ 4 3 and t4 4! eitαn) η 1 it η it)! η 3 + t! η ) : η >ε ) ) ) η I η > ε ) ξ 1 it ξ it)! 4 t 4 ξ 4! 3 η ), we also have ξ ) it)3 3! ) 4 t 4 8 max 1n η ξ ) 3 ) ) ) ) t4 ) 8 max η t4 ) ) 1n 8 max η I η ε + 1n ) + t4 ) ) 8 max η I η > ε 1n t4 8 ε + t4 8 ) ) ) η I η > ε

19 NON LINAR FUNCTIONALS PRSRVING NORMAL DISTRIBUTION AND THIR ASYMPTOTIC NORMALITY 19 Finally we obtain eit η e t / ) ) t 3 6 ε + t4 8 ε + t + t4 ) ) η I η > ε, 8 which completes the proof The authors are grateful to the referee for careful reading of the paper and valuable suggestions References [1] Billingsley P, Convergence of probability measures John Wiley & Sons, New-Yor, 1999 [] Douhan P, Mixing Properties and examples Springer, 1994 [3] Fisher, A Sorohod representation and an invariance principle for sums of weighted iid random variables Rocy Mount J Math, 199, [4] Gnedeno BV, Kolmogorov AN, Limit distributions for sums of independent random variables Addison-Wesley, 1954 [5] Hall P, Heyde CC, Martingale limit theory and its applications Academic Press, New Yor London, 1980 [6] Ibragimov IA, Linni YuV, Independent and stationary sequences of random variables Groningen, Wolters-Noordhoff, 1971 [7] Kevei P, A note on asymptotics on linear combinations of iid random variables Periodica Mathematica Hungarica 60 1), 010, 5 36 [8] Klimov G, Kuzmin A, Probability, Processes, Statistics Problems with solutions in Russian) Moscow University Press, Moscow, 1985 [9] Linni YuV, idlin VL, Remar on analytic transformations of normal vectors Theory of Probability and its Applications 13 4), 1968, [10] Nahapetian BS, Limit theorems and some applications in statistical physics Teubner-Text zur Mathemati 13, 1991

20 0 L A KHACHATRYAN AND B S NAHAPTIAN [11] Petrov VV, Sums of independent random variables Springer, 1975 [1] Shiryaev A, Probability Springer, 1996 [13] Weber M, A weighted central limit theorem Statistics and Probability Letters 76, 006, Linda Khachatryan Institute of mathematics, of National Academy of Sciences of Armenia Bagramian ave 4B, 0019 Yerevan, Armenia linda@instmathsciam Boris S Nahapetian Institute of mathematics, of National Academy of Sciences of Armenia Bagramian ave 4B, 0019 Yerevan, Armenia nahapet@instmathsciam Please, cite to this paper as published in Armen J Math, V 10, N 5018), pp 1 0