Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 2, 2023 21 

 

Georgian Scientists 

ქართველი მეცნიერები 

Vol. 5  Issue 3, 2023 

https://doi.org/10.52340/gs.2023.05.03.03 

 

On the conditional distribution of the sum of m −dependent vectors 

Zurab Kvatadze *, Beqnu Parjiani** and  TSiala Kvatadze *** 

* Professor of the Department of Mathematics  of  Georgian  Technical  University; ** Associate Professor of 

Department of Mathematics  of  Georgian  Technical  University; * Invited Lecturer in Statistics, International 

Black Sea University 

Abstract 

On the  probability space ( , , )F P  is considered a stationary (in the narrow sense) two-

component sequence  
1

,i i i
Y


. : k

iY R   is conditionally m −independent vectors sequence 

The  decomposition  1 1 2

1

1 n

n i n n

i

S Y EY S S
n 

     is applied, where 

 1

1

1 n

n i i i

i

S Y E Y
n




     and  2 1

1

1 n

n i i

i

S E Y EY
n




    . Is proved that if 

2
( ) ( )

nS n
F Q


    and ( )Q   is nondegenerate distribution, then for each , kx y R  

 
1

( ) ( ) ( ) ( ) ( )
m mn n

R R nS
P x y F x x y Q y Q y

 
         , where 

 (0) ( ) ( )

0 0 0

1

m
T

p p

m

p

R R R R


   
   ,    0 1 1 1cov ,

p

p nR E Y Y  ,  1 1 2, ,...,n n     is fixed trajectory of 

control sequence.  When   
1i i




 is a finite ergodic Markov’s chain with one class of ergodicity, 

then is shownthat ( ) ( )TQ R


   , where T   is expressed by chain parameters. When a chain has 

cyclic subclasses, the existence of the corresponding limiting characteristics is understood in the 

sense of Cesaro. 

When solving many practical problems, the question arises of constructing statistical estimates 

of parameters by dependent observations. For example, in a study conducted to determine the 

rating model of vocational colleges, the data obtained from a survey of students are dependent 

on each other, due to the fact that they are obtained from the same social group. (R. Chartolani, 

N. Durglishvili, Z. Kvatadze. Optimization of a State Financing Model of Vocational Colleges. 

Proc. A. Razmadze Math. Inst. 2015. 169.(2-15), pp. 23-31). Also, the dependence of data in 

various studies of the geophysical direction cannot be neglected. For example, in the pre-

earthquake (preparatory) period, tectonic processes in rocks develop continuously in time, and 

therefore records of any characteristic taken at discrete points in time depend on each other as 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 2, 2023 22 

manifestations of one whole. Proceeding from similar considerations, when determining the 

law of distribution of various populations, density estimates are constructed by dependent 

observations. For example, nonparametric density estimates and estimates of regression 

coefficients are known, built by observations connected in a Markov chain  (Yakowitz Sidney 

(1989) Nonparametric density and regression estimation for Markov sequences without mixing 

assumptions. 85721–Journal of Multivariate Analysis, 30: 124-136. Arisona, USA). Conditionally 

independent observations and observations with a chain dependence are also considered. The 

accuracies of density estimates constructed by such observations by the 2L metric (Z, Kvatadze, 

B. Phardjiani. On the Exsactness of Distribution Density Estimates Constructed by Some Class of 

Dependent Observations. Mathematics and Statistics. 2019 Vol. 7(4), pp. 135-145.  SAN JOSE) and by 

the 1L metric (B. Parjiani, L. Labadze, T. Kvatadze; Georgian Scientists, “On the accuracyby the 

metric L1 of the density estimation constructed by dependent observations” Vol. 5. Issue 1, pp. 

308-321, 2023) are known. 

To construct statistical estimates based on dependent observations and determine their 

unbiasedness, it is necessary to know the asymptotic distribution of sums of dependent random 

variables. At the present stage, the rich theory of summation of independent random variables 

(Normal Approximation Some Recent Advances. Sazonov V.V. Lecture Notes in Math. V. 79, 

Berlin, ete.,: Springer. 1981.) is transferred to dependent random variables. In many problems, 

Markov dependence is used, which is one of the types of weak dependence. Questions of the 

limiting asymptotic behavior of sums of random variables of various types of dependency 

(weakly dependent, conditionally independent, connected in a Markov chain) are discussed in 

many papers. Sums of such random variables are often considered, the joint distribution of 

which is determined by some control sequence of random elements. Conditionally independent 

sequences (Bokuchava I. V. Limit theorems for conditionally independent sequences. (in 

Russian) Teor. Verojatnost. i Primenen. XXIX. (1984). №1, p. 192-193) and sequences with chain 

dependence are considered. For example, articles by O'Brien (O’ Braien G.L. Limit Theorems 

for Sum of Chain Dependent Proccesses. U. Appl. Probab., 1974, 11, 582-587), Y. Aleshkevichus 

(G. YU. Aleshkyavichus, On the central limit theorem for sums of random variables given on a 

Markov chain. (Russian) Lithuanian Mathematical Collected Works, Vilnius 6 (1966), №. 1, 

15−22.), R. Chitashvili, T. Shervashidze, I. Bokuchava, Z. Kvatadze (Bokuchava I., Kvatadze Z., 

Shervashidze T. On Limit theorem for random vectors controlled by a Marcov chain. Prob. 

Theory and Math. Stat., Vol. 1, 1986, 239-250. VNU Science Press, Utrecht.) and other authors 

deal with these issues.In this paper, we discuss the class of sequences of conditionally m -

independent vectors and the class of sequences of vectors with chain m -dependence. (Kvatadze 

Z., ShervashidzeT. Some limit theorems for I.I.D. and Conditionally independent random 

variables. The second international Conference, “Problems of Cybernetics and Informatics”. 

September 10-12, 2008. Baku. Azerbaijan. Section № 4. “Applied Stochastic Analysis”. Institute 

of Information Technologies of NASA. Printing House of “Information Technology” Baku. 2008. 

Vol. II, 217-219). In the case of a non-regular ergodic chain, the sequence of functions is defined 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 2, 2023 23 

on a sequence of vectors with a chain m -dependence. A representation in matrix form of the 

limit covariance matrix of the normalized sum of the sequence of these functions is obtained. 

The fundamental matrix and the corresponding limiting characteristics of the Markov chain in 

the case of cyclic subclasses are computed using the Cesaro summation. 

The Levy-Prokhorov metric  , sup ( ) ( )
kx R

F G F x G x


   is often used when considering the 

convergence of distributions on the space of distribution functions. Therefore, the question of 

determining the limiting probability of the distribution of the sum of functions falling into a 

strip consisting of two normal shifted distributions is of natural interest. The article discusses 

the normalized sum of a sequence of conditionally m -dependent vectors. The limiting 

probability of the conditional distribution of this sum falling into the band consisting of two 

shifted normal distributions is determined. The covariance matrix of these distributions is 

expressed by the Markov chain parameters. A similar limiting  probability is set for the 

normalized sum of a sequence of vectors with a chain m -dependence. 

 

Keywords: Conditionally m −independent sequence, conditional distribution,  Markov chain, 

m -dependent sequence controlled by the Markov chain. 

2010 Mathematics Subject Classification.  60F05. 60J10. 60J20. 

  

Introduction 

When solving practical problems, it is often necessary to build statistical estimates based 

on dependent observations. For example, time series used in psychological research or in 

economic parameter estimations, in most cases, consist of dependent data. Determining the 

accuracy and unbiasedness of the constructed estimates is related to the knowledge of the 

asymptotic distribution of the sums of dependent random variables. There is a spread of methods 

for studying the distribution of sums of independent random variables ([1]) over dependent 

random variables. The Markov dependence, which is one of the types of weak dependence, 

naturally enters this path. Limit theorems for weakly dependent sequences are expressed in 

terms of sigma-algebras generated by asymptotically separable segments of the sequence. An 

essential role in their approval is played by Bernstein's "sectioning" method, which uses the 

effect of dependence weakening with increasing distance between segments. Many authors 

study sums of random variables whose distribution is determined by some "control" sequence 

of random elements. Conditionally independent sequences ([2],[3]) and sequences with chain 

dependence ([4],[5]) are considered. In this paper, we consider a conditionally m -independent 

sequence of vectors and a sequence of vectors with a chain m -dependence. Let's calculate the 

limiting probability of  their conditional distribution falling into the band created from two 

shifted normal distributions.  



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 2, 2023 24 

 On the probability space ( , , )F P  we consider a two-component stationary (in the 

narrow sense) sequence  

 
1

,i i i
Y


                                                     (1) 

 
1i i




,  :i    is the control sequence. Assume that the set   consists of a finite number 

of finite elements.  
1i i

Y


,  : k

iY R  is a sequence of conditionally m −independent vectors.   

Definition. (see. [6]) The sequence  
1i i

Y


 from (1) is called a conditionally m

−independent sequence if the  vectors 1 2, , , nY Y Y  on a fixed  trajectory  1 1 2, ,..,n n     for an 

arbitrary natural number n  become independent when thier index difference exceeds  m . In 

this case the  distribution of iY  is depends only on i .  For arbitrary natural numbers  i , l , n , 

1 2, , , lj j j , (2 l n  ; i n ; 1 21 ... lj j j n     ) the equations are fulfilled:  

  1 1 1 2 22
, , ,

... ,        
j j j j j jj j j n l ll

p qY Y YY Y Y
j j m

  
     Tu   ,   , 1,2,...,p q l   

1
1 2,     1 ... ,     1,

i ii n
lYY

j j j n i n


       , 

where X  is the distribution of X . 

  Remark: If  
1i i




 is a Markov chain with discrete time, then the sequence   
1i i

Y


  is 

called a conditionally m -dependent sequence controlled by the Markov chain (sequence with 

chain m -dependence).  

  Let us give auxiliary lemmas. 

Let's introduce notations : 

  1 1( ) ,                ( )j j jE Y E EY         

      1, ,                1 ,
T

j r j j r r nR E Y Y j r n                

   ( ) ( )

0 1 1 0 1 1, ,             , ,                0,...,l l

l lR ER R ER l m   

     

                         ( ) (0) ( ) ( )

0 0 0 0

1

,
m m

T
l l l

m

l m l

R R R R R
 

    
                                     (1) 

Consider sum  
1

1 n

n i

i

S Y
n




  . Let us use the same method that was used to determine 

the limit distribution of the sum nS  by fixing the trajectories 1n  in [5]. Let us decompose the 

sum nS  on a fixed trajectory into two uncorrelated and asymptotically independent centered 

sums 1 2n n nS S S  : 

 1

1

1 n

n j j

j

S Y
n

 


  
  ,               2

1

1 n

n j

i

S
n

  


  
  . 

Lemma 1. (see [5]) Suppose that  
1i i

Y


 is a conditionally m −independent sequence in 

model (1). Let's say for arbitrary function : kR   when  1pE     ( 1,p k , 

   1 2( ), ( ),..., ( )k         ) almost everywhere an convergence is executed    



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 2, 2023 25 

   1

1

1 n
n

j

j

E
n

 



          a. e.                             (2) 

 Let's say  

                                           ( )msp R                                             (3) 

 Then, for n , the following convergences are fulfilled: 

a) 
1 1 mn n

W

RS 
    a. e., 

b) 
1n m

W

S R , 

c) If  
2n

W

S   then  
1n m

W

S R   

Lemma 2. (see. [7]) Let’s say :g T T   is a continuous mapping of a T  metric space 

into a T   metric space. If d

nX X  when n  then ( ) ( )
d

ng X g X .  

Lemma 3. (see. [7]) Assume that, nX , nZ  and X  are random variables with values in a 

separable metric space T . Let's say when n  the limit equalities d

nX X  and 

 , 0n nX Z   are satisfied, then d

nZ X . 

Methodology 

When proving the theorem, we use the methods applicable in article [5] by I. 

Bokuchava, T. Shervashidze and Z. Kvatadze. The sums 1nS  and 2nS  are uncorrelated and 

asymptotically independent (see. [5]). For calculated expressions in inequalities, we use the 

following representation     1nE E E     . This representation allows us to consider the 

terms of these sums on a fixed trajectory as independent quantities. We are investigating an 

ergodic chain that is not regular. Since there are cyclic subclasses, the existence of the 

fundamental matrix of a chain is understood in the sense of Cesaro (see.[8]). In [9], the 

representation of the covariance matrix of the limit distribution of the normalized sum of 

functions defined on a chain was obtained using the characteristics of the chain. 

Main Results  

Theorem 1.  Suppose that  
1i i

Y


 is a conditionally m −dependent sequence  in model (1). 

Suppose the control sequence  
1i i




 satisfies condition (2) and inequality (3) is fulfilled. Then  

the following propositions are true: 

a) Let’s say 
2nS  is weakly converges to some non degenerate distribution Q  with 

distribution function ( )Q  , then for arbitrary vectors ,x y ,  , kx y R  when n  

there is a convergence 

 
1

P ( ) ( ) ( ) ( ) ( )
m mn n

R RS
x y F x x y Q y Q y


                         (4)        

b) The opposite  is true, if (4) is  true, then 
2n

W

S  . 

Proof. Let's use the following representations  



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 2, 2023 26 

   
1 1 1

2P ( ) ( ) ( ) P ( ) ( ) ( )
m m m mn n n n

R R R n RS S
x y F x x y x y F x S x y

 
              = 

   
1 1 1 1

2 2P ( ) ( ) P ( ) ( ) 1
m mn n n n

n R n RS S
F x S x y F x S x y

 
             

   P 0 P 0 1n nM L                                                                                         (5) 

     1 1

2P ( ) ( ) ( ) P 0 P 0 1
m m mR R n R n nx y x S x y M L                          (6) 

where 

1 1
2( ) ( )

mn n
n n RS

M F x S x y


    ; 

1

2( ) ( )
m mn R n RM x S x y     ; 

1 1
2( ) ( )

mn n
n n RS

L F x S x y


    ; 

 1

2( ) ( )
m mn R n RL x S x y     . 

On the space where the sequence (1) is given, let's determine the random variable    

with distribution Q . Due the continuity and monotonicity of the function ( )
mR    

 2P ( ) ( ) ( )
m m mR R n Rx y x S x y          

   2P P ( ) ( )n

ny S y y y Q y Q y            .             (7) 

Let us show that the quantities nM  and nL  have the same limiting distributions as the 

quantities 1

nM  and  1

nL  (respectively). 

 ( )
mR x t   is continuous with respect to t , and 

2n

W

S  , therefore, by virtue of 

Lemma 2, when n , the following convergences are fulfilled 
1

2( ) ( ) ( ) ( )
m m m m

W

n R n R R RM x S x y x x y         , 

1

2( ) ( ) ( ) ( )
m m m m

W

n R n R R RL x S x y x x y         . 

 Let us introduce the distance between the functions ( )f   and ( )g   as following: 

 ( ), ( ) sup ( ) ( )
kx R

f g f x g x


    . 

 It is clear that 

 On a fixed trajectory  1 1 2, ,..,n n    , the sum 2nS  becomes equal to some specific 

(trajectory dependent) number. Therefore, it is clear that the following equations hold 

   1 1,  ,  n n n nM M L L    

1 1 1 1
2 2sup ( ) ( ) sup ( ) ( )

m mn n n n
n R n RS S

k kx R x R

F x S x S F x x
 

 

      . 

 By virtue of Lemma 1, when n  we have  

1 1

sup ( ) ( ) 0
mn n

RS
kx R

F x x




       a. e. 

 Therefore 

 1,  M 0P

n nM  ,         1,  0P

n nL L  . 

 By virtue of Lemma 3, we will have  

( ) ( )
m m

W

n R RM x x y    , 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 2, 2023 27 

( ) ( )
m m

W

n R RL x x y    . 

 We have obtained that the random vectors nM  and nL  have the same limiting 

distributions as the vectors 1

nM  and   1

nL . According to equalities (5) and (6) we will have  

      
1

lim P ( ) ( ) ( ) lim P 0 +P 0 1
m mn n

R R n nSn n
x y F x x y M L

 
            

      1 1

2lim P 0 +P 0 1 lim P ( ) ( ) ( )
m m mn n R R n R

n n
M L x y x S x y

 
               

( ) ( )Q y Q y     

  The proof of point b) is obtained from the fact that the random vectors nM  and nL  

have the same limiting distributions as the vectors 1

nM  and 1

nL  respectively. According to 

equalities (5) and (6) we will have    

 2lim P ( ) ( ) ( )
m m mR R n R

n
x y x S x y


          

         1 1lim P 0 +P 0 1 lim P 0 +P 0 1n n n n
n n

M L M L
 

          

 
1

lim P ( ) ( ) ( ) ( ) ( )
m mn n

R RSn
x y F x x y Q y Q y


          . 

On the other hand 

   2 2P ( ) ( ) ( ) P
m m mR R n R nx y x S x y y S y             

from this we will obtain b). 

 Let’s suppose that  
1n n




 is a finite, homogeneous, stationary, ergodic Markov chain 

with one class of ergodicity, which may contain cyclic subclasses. Let's say the set of states of 

the Markov chain is  1 2, ,..., rb b b  . The limiting stationary distribution is  1 2( , ,..., )r    . 

Consider a function : kf R  defined on a chain that satisfies the conditions of Lemma 

1. In this case, it is known (see [8]) that for the sum 
1

1
[ ( ) ( )]

n

n j j

j

U f Ef
n

 


   when n  

thefollowing convergences take place  

            cov n fU T , 

 
n f

W

U T . 

 Matrix  
, , 1,i jf f

i j k
T t


  elements are expressed by Markov chain parameters  

,

,

( ) ( ) ( )
i j

r

f i jt z z f f       

 

           , , 1,i j k . (8) 

where Z  is the fundamental matrix of the chain 

1

, 1,
1

[ ( )] ( )j

r
j c

Z I P I P z  







 
       

 
 , 

, 1,
,   ,   , 1,

r
r   

    


    , 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 2, 2023 28 

 and  

1 2( , ,..., )kf f f f ,   ( ) ( )i if f b  , 1,i k , 1, r  . 

The symbol  
c

  stands for convergence according to Cesaro. It was shown in [8] that 

when the chain is irregular (contains cyclic subclasses), then both existence Z  and convergence 

 cov n fU T  are understood in the sense of Cesaro. 

In [9], the representation of the matrix fT  is accepted in the matrix form 

[ ( ) ]T T

f dg dg dg dgT F Z Z F       ,                           (9) 

where F  is the following matrix of order k r .  

1 1 1 2 1 1 1 1

2 1 2 2 2 2 2 2

1 2

( ), ( ),..., ( ) (1), (2),..., ( )

( ), ( ),..., ( ) (1), (2),..., ( )

................................... ...............................

( ), ( ),..., ( ) (1)

r

r

k k k r k

f b f b f b f f f r

f b f b f b f f f r
F

f b f b f b f

 
 
  
 
 
  , (2),..., ( )k kf f r

 
 
 
 
 
 

, 

1 2

1 2

1 2

, ,...,

, ,...,
lim( )

...............

, ,...,

r

rn

c
n

r

P

  

  

  



 
 
   
 
 
 

. 

Let's use this fact. Consider, as a function : kf R , the function  

: kR  .        1 2( ), ( ),..., ( )j j j j j j k jf E Y            , 

( ) ( ) ( )p

p j p j j jf E Y     , 1,p k ,  1 2, ,..., k

j j j jY Y Y Y , 

     j jf E Y       , 1, r  , 

( ) ( ) ( )p

p p j jf E Y       , 1,p k ,  1, r  .  

It is clear that  

       
1

r

j j jE E E Y 



      


   . 

Accordingly, as a the sum nU  will be considered  the sum  
2

1

1
( )

n

n j

j

S
n

  


    .   

 We will have the representation obtained from (9) 

[ ( ) ]T T

dg dg dg dgT Z Z           ,      (10) 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 2, 2023 29 

where 

 

1 1 1 2 1 1 1 1

2 1 2 2 2 2 2 2

1 2

( ), ( ),..., ( ) (1), (2),..., ( )

( ), ( ),..., ( ) (1), (2),..., ( )

................................... ...............................

( ), ( ),..., ( ) (1)

r

r

k k k r k

b b b r

b b b r

b b b

     

     


   

 
 
  
 
 
  , (2),..., ( )k k r 

 
 
 
 
 
 

, 

  Accordingly  
, , 1,i j i j k

T t 


 , where  

 
,

,

( ) ( ) ( )
i j

r

i jt z z        

 

             ,  , 1,i j k .    (11) 

And   is the Kronecker symbol. 

Theorem 2.  Suppose that  
1i i

Y


 is a sequence with a chain m -dependence  in model (1).  

Suppose the control sequence  
1i i




 satisfies condition (2) and inequality (3) is fulfilled. Suppose 

 
1n n




 is a finite, homogeneous, stationary, ergodic Markov chain with one ergodicity class, 

which may contain cyclic subclasses. Let's say the set of states of the Markov chain is 

 1 2, ,..., rb b b  . The limiting stationary distribution is 1 2( , ,..., )r    . Then in the above 

notation for any vectors  , kx y R  when n  occurs the convergence ,x y     

   
1

P ( ) ( ) ( ) ( ) ( )
m mn n

R R T TS
x y F x x y y y

 
              (12) 

Proof. The proof of the theorem is carried out similarly to Theorem 1. It is only necessary to 

take into account that the conditions of Theorem 2 include the conditions of Theorem 1. At the 

moment, it is known that the limit distribution of the sum 2nS  is expressed by the parameters 

of the Markov chain.  In particular, its limiting covariance matrix is established. It can be seen 

from (10) that at n   

2n

W

S T
 . 

 This means that instead of a distribution of ( )Q  , we can consider a distribution of ( )T
 

. The form of the matrix T  in the presence of cyclic subclasses is obtained with the help of 

Cesaro convergence. 

 Discussion   

In the particular case when the Markov chain is regular (does not contain cyclic 

subclasses) and  
1n n

Y


 are one-dimensional ( 1k  ) discrete random variables, then an analogue 

of (11) was obtained in [10]. From here we can obtain (11) by the Cramer-Wold method. 

However, this does not give us the matrix representation (10). In [8], the case 1k  was 

considered, when the Markov chain already contains cyclic subclasses. In [9], a matrix 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 2, 2023 30 

representation (9) was obtained in the general case when 1k   and the chain has cyclic 

subclasses. In this case, as in [8], the result is obtained using Cesàro convergence. An analog of 

Theorem 2 was obtained in [11] for one-dimensional integer random variables connected by a 

regular Markov chain. This result was generalized in [12] by partially removing the restriction 

on random variables. Conditionally m -independent and chain m -dependent one-dimensional 

random variables are discussed in [12]. The case when the chain has no cyclic subclasses is 

considered. Analogues (4) and (12) are obtained for this particular case. 

Conclusion. 

In the work, we have established the probability of the conditional distributions of 

partial sums of a sequence of conditionally m-independent of random vectors falling into a strip 

consisting of two shifted normal distributions. From this theorem, a similar result is obtained 

for sequences of random vectors with a chain m −dependence. This latter is a generalization of 

the result obtained by Z. Bezhaeva [11] to the multidimensional case, when the chain is not 

regular and has cyclic subclasses. A matrix representation of the covariance matrix of this shifted 

normal distribution with characteristics of a Markov chain is given. These characteristics are 

established in the sense of convergence and summation according to Cesàro. 

 

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Baku. 2008. Vol. II, 217-219., 

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11, 582-587. https://www.cambridge.org/core/journals/journal-of-applied-

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5. Bokuchava I., Kvatadze Z., Shervashidze T. On Limit theorem for random vectors controlled by a Marcov 

chain. Prob. Theory and Math. Stat., Vol. 1, 1986, 231-250. VNU Science Press, Utrecht. 

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6. Kvatadze Z., Pharjiani B., Kvatadze TS., One Example  of   m − Dependent Vector’s Sequence.  Report 

of XXXV enlarged session of the seminar of Ilia Vekua Institute of Applied Mathematics (VIAM), of 

Ivane Javakhisvili Tbilisi State University (TSU). Sept. 16-19., 2021; Volume, 35. pp. 51-54. 

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Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 2, 2023 32 

m  დამოკიდებულ ვექტორთა ჯამის პირობითი განაწილების შესახებ 

ზურაბ ქვათაძე*, ბექნუ ფარჯიანი**, ციალა ქვათაძე*** 

* სტუ-ს მათემატიკის დეპარტამენტი. პროფესორი,  ** სტუ-ს მათემატიკის დეპარტამენტი. 

ასოცირებული პროფესორი,  *** სტატისტიკის ლექტორი, შავი სღვის საერთაშორისო უნივერსიტეტი, 

  აბსტრაქტი 

მრავალი პრაქტიკული ამოცანის გადაჭრის დროს დგება დამოკიდებული 

დაკვირვებებით პარამეტრების სტატისტიკური შეფასებების აგების საკითხი. მაგალითად 

პროფესიული კოლეჯების რანჟირების მოდელის დასადგენად ჩატარებულ კვლევაში 

სტუდენტების გამოკითხვით მიღებული მონაცემები დამოკიდებულია ერთმანეთზე, 

გამომდინარე იქედან, რომ ისინი მიღებულია ერთი სოციუმის ჯგუფიდან (R. Chartolani, N. 

Durglishvili, Z. Kvatadze. Optimization of a State Financing Model of Vocational Colleges. Proc. A. 

Razmadze Math. Inst. 2015. 169.(2-15), pp. 23-31). ასევე ვერ უგულვებელვყოფთ მონაცემთა 

დამოკიდებულებას გეოფიზიკური  მიმართულების სხვადასხვა კვლევების დროსაც. 

მაგალითად მიწისძვრის წინა (მოსამზადებელ) პერიოდში ქანებში მიმდინარე ტექტონიკური 

პროცესები დროში უწყვეტად ვითარდება და შესაბამისად დისკრეტულ მომენტებში 

აღებული ნებისმიერი მახასიათებლის ჩანაწერები როგორც ერთი მთლიანის გამოვლინებები 

დამოკიდებულია ერთმანეთზე. ანალოგიური მოსაზრებებიდან გამომდინარე სხვადასხვა 

პოპულაციების განაწილების კანონის დადგენის დროს ხდება სიმკვრივის შეფასების აგება 

დამოკიდებული დაკვირვებებით. მაგალითად ცნობილია სიმკვრივის არაპარამეტრული 

შეფასებები და რეგრესიის კოეფიციენტების შეფასებები რომლებიც აგებულია მარკოვის 

ჯაჭვად შეკრული დაკვირვებებით (Yakowitz Sidney (1989) Nonparametric density and regression 

estimation for Markov sequences without mixing assumptions. 85721–Journal of Multivariate 

Analysis, 30: 124-136. Arisona, USA). ასევე განიხილება პირობითად დამოუკიდებელი  და 

ჯაჭვურად დამოკიდებული  დაკვირვებები. ცნობილია მათი საშუალებით აგებული 

სიმკვრივის გულოვანი შეფასებები და მათი სიზუსტე 2L  მეტრიკით (Z, Kvatadze, B. Phardjiani. 

On the Exsactness of Distribution Density Estimates Constructed by Some Class of Dependent Observations. 

Mathematics and Statistics. 2019 Vol. 7(4), pp. 135-145.  SAN JOSE) და 1L  მეტრიკით  (B. Parjiani, L. 

Labadze, T. Kvatadze; Georgian Scientists, “On the accuracyby the metric L1 of the density estimation 

constructed by dependent observations” Vol. 5.   Issue 1, pp. 308-321, 2023). 

დამოკიდებული დაკვირვებებით სტატისტიკური შეფასებების ასაგებად და მათი 

ძალმოსილობის დასადგენად საჭიროა დამოკიდებელი შემთხვევითი სიდიდეების ჯამების 

განაწილების ასიმპტოტიკის ცოდნა.  თანამედროვე ეტაპზე ხდება დამოუკიდებელ 

შემთხვევით სიდიდეთა შეკრების მდიდარი თეორიის (Normal Approximation Some Recent 

Advances. Sazonov V. V. Lecture Notes in Math. V. 79, Berlin, ete., :Springer. 1981.) გადატანა 

დამოკიდებულ შემთხვევით სიდიდეებზე. მრავალ ამოცანაში გამოიყენება მარკოვული 

დამოკიდებულება, რომელიც სუსტად დამოკიდებულების ერთ-ერთი სახეა. სხვადასხვა 

ტიპის დამოკიდებულების (სუსტად დამოკიდებული, პირობითად დამოუკიდებული, 



Georgian Scientists/ქართველი მეცნიერები ტ. 5  N 2, 2023 33 

მარკოვის ჯაჭვად შეკრული) შემთხვევითი სიდიდეების ჯამების ზღვარითი ასიმპტოტიკის 

საკითხები მრავალ ნაშრომშია გადმოცემული. ხშირად განიხილება ისეთ შემთხვევით 

სიდიდეთა ჯამები, რომელთა ერთობლივი განაწილება განისაზღვრება შემთხვევით 

ელემენტთა რაიმე მმართველი მიმდევრობით. განიხილება პირობითად დამოუკიდებელი 

(Bokuchava I. V. Limit theorems for conditionally independent sequences. (in Russian) Teor. 

Verojatnost. i Primenen. XXIX. (1984). №1, p. 192-193) და ჯაჭვურად დამოკიდებული 

მიმდევრობები. ამ საკითხებს ეხება მაგალითად გ. ობრაინის ( O’ Braien G.L. Limit Theorems 

for Sum of Chain Dependent Proccesses. U. Appl. Probab., 1974, 11, 582-587); ი. ალეშკიავიჩუსის ( 

G. YU. Aleshkyavichus, On the central limit theorem for sums of random variables given on a Markov 

chain. (Russian) Lithuanian Mathematical Collected Works, Vilnius 6 (1966), №. 1, 15−22.); რ. 

ჩიტაშვილის, თ. შერვაშიძის, ი. ბოკუჩავას, ზ. ქვათაძის (Bokuchava I., Kvatadze Z., Shervashidze 

T. On Limit theorem for random vectors controlled by a Marcov chain. Prob. Theory and Math. Stat., 

Vol. 1, 1986, 239-250. VNU Science Press, Utrecht.)  და სხვათა ნაშრომები.   წინამდებარე  

ნაშრომში განხილულია პირობითად m −დამოუკიდებულ ვექტორთა მიმდევრობების კლასი 

და ჯაჭვურად m −დამოკიდებულ ვექტორთა მიმდევრობების კლასი (Kvatadze Z., 

ShervashidzeT. Some limit theorems for I.I.D. and Conditionally independent random variables. The 

second international Conference, “Problems of Cybernetics and Informatics”. September 10-12, 2008. 

Baku. Azerbaijan. Section № 4. “Applied Stochastic Analysis”. Institute of Information Technologies of 

NASA. Printing House of “Information Technology” Baku. 2008. Vol. II, 217-219). დადგენილია 

მათი ნორმირებული ჯამის ზღვარითი განაწილება. არარეგულარული ერგოდული ჯაჭვის 

შემთხვევაში  ჯაჭვურად m −დამოკიდებულ ვექტორთა მიმდევრობაზე განსაზღვრულია 

ფუნქციათა მიმდევრობა. მიღებულია ამ ფუნქციათა მიმდევრობის ნორმირებული ჯამის 

ზღვარითი კოვარიაციის მატრიცის წარმოდგენა მატრიცული სახით. ჯაჭვის შესაბამისი 

ფუნდამენტური მატრიცა და ზღვარითი მახასიათებლები ციკლური ქვეკლასების არსებობის 

შემთხვევაში გამოთვლილია ჩეზაროს აზრით კრებადობის გამოყენებით.  

განაწილების ფუნქციათა სივრცეზე განაწილებების კრებადობის განხილვის დროს 

ხშირად გამოიყენება ლევი-პროხოროვის მეტრიკა  , sup ( ) ( )
kx R

F G F x G x


  . ამიტომ 

ბუნებრივია საინტერესოა საკითხი ფუნქციათა ჯამის განაწილების ორი ნორმალური 

წანაცვლებული განაწილებისგან შედგენილ ზოლში მოხვედრის ზღვარითი ალბათობის 

დადგენის შესახებ. ნაშრომში განხილულია პირობითად m −დამოკიდებულ ვექტორთა 

მიმდევრობის ნორმირებული ჯამი. დადგენილია ამ ჯამის პირობითი განაწილების ორი 

წანაცვლებული ნორმალური განაწილებისგან შედგენილ ზოლში მოხვედრის ზღვარითი 

ალბათობა. ამ განაწილებების კოვარიაციის მატრიცა გამოსახულია ჯაჭვის პარამეტრების 

საშუალებით. დადგენილია ასეთივე ზღვარითი ალბათობა ჯაჭვურად m −დამოკიდებულ 

ვექტორთა მიმდევრობის ნორმირებული ჯამისთვის.