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A discrete probability distribution concentrated on a set of points of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l0576401.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l0576402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l0576403.png" /> is a real number and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l0576404.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l0576405.png" /> is called the step of the lattice distribution, and if for no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l0576406.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l0576407.png" /> the distribution is concentrated on a set of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l0576408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l0576409.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764010.png" /> is called the maximal step. An [[Arithmetic distribution|arithmetic distribution]] is a particular case (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764011.png" />) of a lattice distribution.
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For a probability distribution with characteristic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764012.png" /> to be a lattice distribution it is necessary and sufficient that there exists a real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764014.png" />; in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764015.png" /> is the maximal step if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764016.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764018.png" />. The characteristic function of a lattice distribution is periodic.
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A discrete probability distribution concentrated on a set of points of the form  $  a + nh $,
 +
where  $  h > 0 $,
 +
$  a $
 +
is a real number and  $  n = 0, \pm  1, \pm  2 , .  . . $.  
 +
The number  $  h $
 +
is called the step of the lattice distribution, and if for no  $  a _ {1} $
 +
and  $  h _ {1} > h $
 +
the distribution is concentrated on a set of the form  $  a _ {1} + nh _ {1} $,
 +
$  n = 0, \pm  1 , \pm  2 \dots $
 +
then  $  h $
 +
is called the maximal step. An [[Arithmetic distribution|arithmetic distribution]] is a particular case ( $  a= 0 $)
 +
of a lattice distribution.
 +
 
 +
For a probability distribution with characteristic function  $  f( t) $
 +
to be a lattice distribution it is necessary and sufficient that there exists a real number $  t _ {0} \neq 0 $
 +
such that $  | f( t _ {0} ) | = 1 $;  
 +
in this case $  h $
 +
is the maximal step if and only if $  | f( t) | < 1 $
 +
for  $  0 < t < 2 \pi /h $
 +
and  $  | f( 2 \pi /h) | = 1 $.  
 +
The characteristic function of a lattice distribution is periodic.
  
 
The inversion formula for a lattice distribution has the form
 
The inversion formula for a lattice distribution has the form
  
<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/l/l057/l057640/l05764019.png" /></td> </tr></table>
+
$$
 +
p _ {n}  =
 +
\frac{h}{2 \pi }
 +
\int\limits _ {| t | < \pi /h } e ^ {- it( a+ nh) } f( t)  dt ,
 +
$$
 +
 
 +
where  $  p _ {n} $
 +
is the probability that the lattice distribution ascribes to the point  $  a + nh $
 +
and  $  f( t) $
 +
is the corresponding characteristic function. The following equality also holds:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764020.png" /> is the probability that the lattice distribution ascribes to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764022.png" /> is the corresponding characteristic function. The following equality also holds:
+
$$
 +
\sum _ {n = - \infty } ^  \infty  p _ {n}  ^ {2}  = \
  
<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/l/l057/l057640/l05764023.png" /></td> </tr></table>
+
\frac{h}{2 \pi }
 +
\int\limits _ {| t | < \pi /h } | f( t) |  ^ {2}  dt.
 +
$$
  
The convolution of two lattice distributions with steps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764025.png" /> and with finite supports is a lattice distribution if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764026.png" /> is a rational number.
+
The convolution of two lattice distributions with steps $  h _ {1} $
 +
and $  h _ {2} $
 +
and with finite supports is a lattice distribution if and only if $  h _ {1} /h _ {2} $
 +
is a rational number.
  
In the study of the limit behaviour of sums of independent random variables, the basic result of the [[Central limit theorem|central limit theorem]] on convergence towards the normal distribution is considerably complemented by local theorems for lattice distributions. The simplest example of a local theorem for lattice distributions is the [[Laplace theorem|Laplace theorem]], which can be generalized as follows: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764027.png" /> be a sequence of independent identically-distributed random variables with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764029.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764030.png" /> while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764031.png" /> takes values of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764033.png" />. Put
+
In the study of the limit behaviour of sums of independent random variables, the basic result of the [[Central limit theorem|central limit theorem]] on convergence towards the normal distribution is considerably complemented by local theorems for lattice distributions. The simplest example of a local theorem for lattice distributions is the [[Laplace theorem|Laplace theorem]], which can be generalized as follows: Let $  X _ {1} , X _ {2} \dots $
 +
be a sequence of independent identically-distributed random variables with $  {\mathsf E} X _ {1} = m $,  
 +
$  {\mathsf D} X _ {1} = \sigma  ^ {2} $,  
 +
and let $  S _ {k} = X _ {1} + \dots + X _ {k} $
 +
while $  X _ {1} $
 +
takes values of the form $  a + nh $,
 +
$  h > 0 $.  
 +
Put
  
<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/l/l057/l057640/l05764034.png" /></td> </tr></table>
+
$$
 +
P _ {k} ( n)  = {\mathsf P} \{ S _ {k} = ka + nh \} ;
 +
$$
  
 
for the asymptotic relation
 
for the asymptotic relation
  
<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/l/l057/l057640/l05764035.png" /></td> </tr></table>
+
$$
 +
\left |
 +
\frac{\sigma \sqrt k }{h}
 +
P _ {k} ( n) -  
 +
\frac{1}{\sqrt {2 \pi } }
 +
  \mathop{\rm exp} \left \{
 +
-
 +
\frac{1}{2}
 +
\left (
 +
\frac{ka + nh - km }{\sigma \sqrt h }
 +
\right )  ^ {2} \right \} \
 +
\right |  \rightarrow  0,
 +
$$
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764036.png" />, to be true uniformly with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764037.png" />, it is necessary and sufficient that the step <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057640/l05764038.png" /> is the maximal step.
+
as $  k \rightarrow \infty $,  
 +
to be true uniformly with respect to $  n $,  
 +
it is necessary and sufficient that the step $  h $
 +
is the maximal step.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Gnedenko,  "The theory of probability" , Chelsea, reprint  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Petrov,  "Sums of independent random variables" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</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></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Gnedenko,  [[Gnedenko, "A course in the theory of probability"|"The theory of probability"]], Chelsea, reprint  (1962)  (Translated from Russian)</TD></TR>
 
+
<TR><TD valign="top">[2]</TD> <TD valign="top">  V.V. Petrov,  "Sums of independent random variables" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</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></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1–2''' , Wiley  (1957–1971)</TD></TR>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1–2''' , Wiley  (1957–1971)</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Lukacs,  "Characteristic functions" , Griffin  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.L. Johnson,  S. Kotz,  "Distributions in statistics: discrete distributions" , Mifflin  (1969)</TD></TR></table>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Lukacs,  "Characteristic functions" , Griffin  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.L. Johnson,  S. Kotz,  "Distributions in statistics: discrete distributions" , Mifflin  (1969)</TD></TR></table>

Latest revision as of 22:15, 5 June 2020


A discrete probability distribution concentrated on a set of points of the form $ a + nh $, where $ h > 0 $, $ a $ is a real number and $ n = 0, \pm 1, \pm 2 , . . . $. The number $ h $ is called the step of the lattice distribution, and if for no $ a _ {1} $ and $ h _ {1} > h $ the distribution is concentrated on a set of the form $ a _ {1} + nh _ {1} $, $ n = 0, \pm 1 , \pm 2 \dots $ then $ h $ is called the maximal step. An arithmetic distribution is a particular case ( $ a= 0 $) of a lattice distribution.

For a probability distribution with characteristic function $ f( t) $ to be a lattice distribution it is necessary and sufficient that there exists a real number $ t _ {0} \neq 0 $ such that $ | f( t _ {0} ) | = 1 $; in this case $ h $ is the maximal step if and only if $ | f( t) | < 1 $ for $ 0 < t < 2 \pi /h $ and $ | f( 2 \pi /h) | = 1 $. The characteristic function of a lattice distribution is periodic.

The inversion formula for a lattice distribution has the form

$$ p _ {n} = \frac{h}{2 \pi } \int\limits _ {| t | < \pi /h } e ^ {- it( a+ nh) } f( t) dt , $$

where $ p _ {n} $ is the probability that the lattice distribution ascribes to the point $ a + nh $ and $ f( t) $ is the corresponding characteristic function. The following equality also holds:

$$ \sum _ {n = - \infty } ^ \infty p _ {n} ^ {2} = \ \frac{h}{2 \pi } \int\limits _ {| t | < \pi /h } | f( t) | ^ {2} dt. $$

The convolution of two lattice distributions with steps $ h _ {1} $ and $ h _ {2} $ and with finite supports is a lattice distribution if and only if $ h _ {1} /h _ {2} $ is a rational number.

In the study of the limit behaviour of sums of independent random variables, the basic result of the central limit theorem on convergence towards the normal distribution is considerably complemented by local theorems for lattice distributions. The simplest example of a local theorem for lattice distributions is the Laplace theorem, which can be generalized as follows: Let $ X _ {1} , X _ {2} \dots $ be a sequence of independent identically-distributed random variables with $ {\mathsf E} X _ {1} = m $, $ {\mathsf D} X _ {1} = \sigma ^ {2} $, and let $ S _ {k} = X _ {1} + \dots + X _ {k} $ while $ X _ {1} $ takes values of the form $ a + nh $, $ h > 0 $. Put

$$ P _ {k} ( n) = {\mathsf P} \{ S _ {k} = ka + nh \} ; $$

for the asymptotic relation

$$ \left | \frac{\sigma \sqrt k }{h} P _ {k} ( n) - \frac{1}{\sqrt {2 \pi } } \mathop{\rm exp} \left \{ - \frac{1}{2} \left ( \frac{ka + nh - km }{\sigma \sqrt h } \right ) ^ {2} \right \} \ \right | \rightarrow 0, $$

as $ k \rightarrow \infty $, to be true uniformly with respect to $ n $, it is necessary and sufficient that the step $ h $ is the maximal step.

References

[1] B.V. Gnedenko, "The theory of probability", Chelsea, reprint (1962) (Translated from Russian)
[2] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)
[3] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)

Comments

References

[a1] W. Feller, "An introduction to probability theory and its applications", 1–2 , Wiley (1957–1971)
[a2] E. Lukacs, "Characteristic functions" , Griffin (1970)
[a3] N.L. Johnson, S. Kotz, "Distributions in statistics: discrete distributions" , Mifflin (1969)
How to Cite This Entry:
Lattice distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lattice_distribution&oldid=25523
This article was adapted from an original article by N.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article