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''series scheme, serial scheme, triangular array''
 
''series scheme, serial scheme, triangular array''
  
A double sequence of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s0845701.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s0845702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s0845703.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s0845704.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s0845705.png" /> (forming the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s0845706.png" />-th series) are mutually independent for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s0845707.png" />. The simplest sequence of series corresponds to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s0845708.png" />. In the limit theorems of probability theory, a fundamental problem is to determine the limiting behaviour, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s0845709.png" />, of the distributions of the random variables
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\eta _ {n}  = \
 +
\sum _ { k = 1 } ^ { {k _ n } }
 +
\xi _ {nk} .
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457011.png" /> satisfies the condition of uniform asymptotic negligibility, that is,
+
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,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\max _ {1 \leq  k \leq  k _ {n} } \
 +
{\mathsf P} \{ | \xi _ {nk} | \geq  \epsilon \}  \rightarrow  0
 +
$$
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457013.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457014.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457015.png" /> is a null sequence of series satisfying the condition of uniform asymptotic negligibility, coincides with the set of infinitely-divisible distributions.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457016.png" /> 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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457017.png" /> be a sequence of series, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457018.png" /> be the distribution function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457019.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457020.png" /> to satisfy (2), and for the distribution of the sums (2) to be weakly convergent to the normal distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457022.png" />, it is necessary and sufficient that for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457023.png" /> the following conditions hold:
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457024.png" />,
+
1) $  \sum _ {k = 1 }  ^ {k _ {n} } {\mathsf P} \{ | \xi _ {nk} | \geq  \epsilon \} \rightarrow 0 $,
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457025.png" />,
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457026.png" />.
+
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.
 
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.
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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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457027.png" /></td> </tr></table>
+
$$
 +
\xi _ {nk}  = \
 +
 
 +
\frac{\xi _ {k} - {\mathsf E} \xi _ {k} }{\sqrt {\sum _ {k = 1 } ^ { n }  {\mathsf D} \xi _ {k} } }
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457028.png" /></td> </tr></table>
+
$$
 +
\xi _ {nk}  =
 +
\frac{\xi _ {k} - {\mathsf E} \xi _ {k} }{n}
 +
,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084570/s08457029.png" /> are independent random variables.
+
where the $  \xi _ {k} $
 +
are independent random variables.
  
 
====References====
 
====References====

Latest revision as of 08:13, 6 June 2020


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). 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 [1]). In particular, the condition for convergence to a 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; Law of large numbers; and 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

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

Comments

References

[a1] M. Loève, "Probability theory" , I , Springer (1977)
How to Cite This Entry:
Sequence of series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sequence_of_series&oldid=40031
This article was adapted from an original article by N.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article