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''series scheme, serial scheme, triangular array''

A double sequence of random variables  $  \xi _ {nk} $, 
$  1 \leq  k \leq  k _ {n} $, 
$  k _ {n} \rightarrow \infty $, 
$  n \geq  1 $, 
in which  $  \xi _ {n1} \dots \xi _ {nk _ {n}  } $(
forming the  $  n $-
th series) are mutually independent for any  $  n $. 
The simplest sequence of series corresponds to the case  $  k _ {n} = n $. 
In the limit theorems of probability theory, a fundamental problem is to determine the limiting behaviour, as  $  n \rightarrow \infty $, 
of the distributions of the random variables

$$ \tag{1 }
\eta _ {n}  = \ 
\sum _ { k = 1 } ^ { {k _ n } }
\xi _ {nk} .
$$

Under suitable conditions the class of limit distributions for such sequences coincides with the class of infinitely-divisible distributions (cf. [[Infinitely-divisible distribution|Infinitely-divisible distribution]]). Suppose that the sequence of series  $  \xi _ {nk} $
satisfies the condition of uniform asymptotic negligibility, that is,

$$ \tag{2 }
\max _ {1 \leq  k \leq  k _ {n} } \ 
{\mathsf P} \{ | \xi _ {nk} | \geq  \epsilon \}  \rightarrow  0
$$

as  $  n \rightarrow \infty $. 
Then  $  \xi _ {nk} $
is said to form a null sequence of series. The set of distributions that are the limit distributions (in the sense of weak convergence) for the random variables (1), where  $  \xi _ {nk} $
is a null sequence of series satisfying the condition of uniform asymptotic negligibility, coincides with the set of infinitely-divisible distributions.

There are conditions for the convergence of the distributions of  $  \eta _ {n} $
to a given infinitely-divisible distribution (see [[#References|[1]]]). In particular, the condition for convergence to a [[Normal distribution|normal distribution]] has the following form.

Let  $  \xi _ {nk} $
be a sequence of series, and let  $  F _ {nk} $
be the distribution function of  $  \xi _ {nk} $. 
For  $  \xi _ {nk} $
to satisfy (2), and for the distribution of the sums (2) to be weakly convergent to the normal distribution with parameters  $  a $
and  $  b $, 
it is necessary and sufficient that for any fixed  $  \epsilon > 0 $
the following conditions hold:

1)  $  \sum _ {k = 1 }  ^ {k _ {n} } {\mathsf P} \{ | \xi _ {nk} | \geq  \epsilon \} \rightarrow 0 $,

2)  $  \sum _ {k = 1 }  ^ {k _ {n} } \left \{ \int _ {| x | < \epsilon }  x  ^ {2}  dF _ {nk} ( x) - \left ( \int _ {| x | < \epsilon }  x  dF _ {nk} ( x) \right )  ^ {2} \right \} \rightarrow b  ^ {2} $,

3)  $  \sum _ {k = 1 }  ^ {k _ {n} } \int _ {| x | < \epsilon }  x  dF _ {nk} ( x) \rightarrow a $.

The study of limit distributions for the normalized partial sums of a sequence of independent random variables is a special case of the study of sequences of series.

As regards sequences of series see also: [[Infinitely-divisible distribution|Infinitely-divisible distribution]]; [[Law of large numbers|Law of large numbers]]; and [[Limit theorems|Limit theorems]]. For example, in classical versions of the central limit theorem and the law of large numbers, one considers special cases of sequences of series, formed by random variables

$$ 
\xi _ {nk}  = \ 

\frac{\xi _ {k} - {\mathsf E} \xi _ {k} }{\sqrt {\sum _ {k = 1 } ^ { n }  {\mathsf D} \xi _ {k} } }
 ,
$$

$$ 
\xi _ {nk}  =  
\frac{\xi _ {k} - {\mathsf E} \xi _ {k} }{n}
 ,
$$

where the  $  \xi _ {k} $
are independent random variables.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables", Addison-Wesley (1954) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes", Springer (1969) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.V. Petrov, "Sums of independent random variables", Springer (1975) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1971) pp. Chapt. 1</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Loève,   "Probability theory" , '''I''' , Springer  (1977)</TD></TR></table>