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[[Category:Stochastic processes]]
 
[[Category:Stochastic processes]]
  
A formula for the logarithm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l0582701.png" /> of the [[Characteristic function|characteristic function]] of an [[Infinitely-divisible distribution|infinitely-divisible distribution]]:
+
A formula for the logarithm $  \mathop{\rm ln}  \phi ( \lambda ) $
 +
of the [[Characteristic function|characteristic function]] of an [[Infinitely-divisible distribution|infinitely-divisible distribution]]:
  
<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/l058/l058270/l0582702.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm ln}  \phi ( \lambda )  = i \gamma \lambda -
  
<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/l058/l058270/l0582703.png" /></td> </tr></table>
+
\frac{\sigma  ^ {2} \lambda  ^ {2} }{2}
 +
+
 +
\int\limits _ {- \infty } ^ { 0 }
 +
\left (
 +
e ^ {i \lambda x } - 1 -
  
where the characteristics of the Lévy canonical representation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l0582704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l0582705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l0582706.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l0582707.png" />, satisfy the following conditions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l0582708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l0582709.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827011.png" /> are non-decreasing left-continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827013.png" />, respectively, such that
+
\frac{i \lambda x }{1 + x  ^ {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/l058/l058270/l05827014.png" /></td> </tr></table>
+
\right ) \
 +
d M ( x) +
 +
$$
 +
 
 +
$$
 +
+
 +
\int\limits _ { 0 } ^  \infty  \left ( e ^ {i \lambda x } - 1
 +
-
 +
\frac{i \lambda x }{1 + x  ^ {2} }
 +
\right )  d N ( x) ,
 +
$$
 +
 
 +
where the characteristics of the Lévy canonical representation,  $  \gamma $,
 +
$  \sigma  ^ {2} $,
 +
$  M $,
 +
and  $  N $,
 +
satisfy the following conditions: $  - \infty < \gamma < \infty $,
 +
$  \sigma  ^ {2} \geq  0 $,
 +
and  $  M ( x) $
 +
and  $  N ( x) $
 +
are non-decreasing left-continuous functions on  $  ( - \infty , 0 ) $
 +
and  $  ( 0 , \infty ) $,
 +
respectively, such that
 +
 
 +
$$
 +
\lim\limits _ {x \rightarrow \infty } \
 +
N ( x)  = \lim\limits _ {x \rightarrow - \infty } \
 +
M ( x)  = 0
 +
$$
  
 
and
 
and
  
<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/l058/l058270/l05827015.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { - 1} ^ { 0 }  x  ^ {2}  d M ( x)  < \infty ,\ \
 +
\int\limits _ { 0 } ^ { 1 }  x  ^ {2}  d N ( x)  <  \infty .
 +
$$
  
To every infinitely-divisible distribution there corresponds a unique system of characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827019.png" /> in the Lévy canonical representation, and conversely, under the above conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827023.png" /> the Lévy canonical representation with respect to such a system determines the logarithm of the characteristic function of some infinitely-divisible distribution.
+
To every infinitely-divisible distribution there corresponds a unique system of characteristics $  \gamma $,  
 +
$  \sigma  ^ {2} $,  
 +
$  M $,  
 +
$  N $
 +
in the Lévy canonical representation, and conversely, under the above conditions on $  \gamma $,  
 +
$  \sigma  ^ {2} $,  
 +
$  M $,  
 +
and $  N $
 +
the Lévy canonical representation with respect to such a system determines the logarithm of the characteristic function of some infinitely-divisible distribution.
  
Thus, for the [[Normal distribution|normal distribution]] with mean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827024.png" /> and variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827025.png" />:
+
Thus, for the [[Normal distribution|normal distribution]] with mean $  a $
 +
and variance $  \sigma  ^ {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/l058/l058270/l05827026.png" /></td> </tr></table>
+
$$
 +
\gamma  = a ,\  \sigma  ^ {2}  = \sigma  ^ {2} ,\ \
 +
N ( x)  \equiv  0 ,\  M ( x)  \equiv  0 .
 +
$$
  
For the [[Poisson distribution|Poisson distribution]] with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827027.png" />:
+
For the [[Poisson distribution|Poisson distribution]] with parameter $  \lambda $:
  
<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/l058/l058270/l05827028.png" /></td> </tr></table>
+
$$
 +
\gamma  =
 +
\frac \lambda {2}
 +
,\ \
 +
\sigma  ^ {2}  = 0 ,\ \
 +
M ( x)  \equiv  0 ,\ \
 +
N ( x)  = \left \{
 +
\begin{array}{rl}
 +
- \lambda  & \textrm{ for }  x \leq  1 ,  \\
 +
0  & \textrm{ for }  x > 1 . \\
 +
\end{array}
  
To the [[Stable distribution|stable distribution]] with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827030.png" />, corresponds the Lévy representation with
+
\right .$$
  
<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/l058/l058270/l05827031.png" /></td> </tr></table>
+
To the [[Stable distribution|stable distribution]] with exponent  $  \alpha $,
 +
0 < \alpha < 2 $,
 +
corresponds the Lévy representation with
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827033.png" />, are constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827034.png" />. The Lévy canonical representation of an infinitely-divisible distribution was proposed by P. Lévy in 1934. It is a generalization of a formula found by A.N. Kolmogorov in 1932 for the case when the infinitely-divisible distribution has finite variance. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827035.png" /> there is a formula equivalent to the Lévy canonical representation, proposed in 1937 by A.Ya. Khinchin and called the [[Lévy–Khinchin canonical representation|Lévy–Khinchin canonical representation]]. The probabilistic meaning of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827037.png" /> and the range of applicability of the Lévy canonical representation are defined as follows: To every infinitely-divisible distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827038.png" /> corresponds a stochastically-continuous process with stationary independent increments
+
$$
 +
\sigma  ^ {2}  = 0 ,\ \
 +
\textrm{ any } \
 +
\gamma ,\  M ( x)  =
 +
\frac{c _ {1} }{| x |  ^  \alpha  }
 +
,\ \
 +
N ( x)  = -  
 +
\frac{c _ {2} }{x  ^  \alpha  }
 +
,
 +
$$
  
<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/l058/l058270/l05827039.png" /></td> </tr></table>
+
where  $  c _ {i} \geq  0 $,
 +
$  i = 1 , 2 $,
 +
are constants  $  ( c _ {1} + c _ {2} > 0 ) $.
 +
The Lévy canonical representation of an infinitely-divisible distribution was proposed by P. Lévy in 1934. It is a generalization of a formula found by A.N. Kolmogorov in 1932 for the case when the infinitely-divisible distribution has finite variance. For  $  \mathop{\rm ln}  \phi ( \lambda ) $
 +
there is a formula equivalent to the Lévy canonical representation, proposed in 1937 by A.Ya. Khinchin and called the [[Lévy–Khinchin canonical representation|Lévy–Khinchin canonical representation]]. The probabilistic meaning of the functions  $  N $
 +
and  $  M $
 +
and the range of applicability of the Lévy canonical representation are defined as follows: To every infinitely-divisible distribution function  $  F $
 +
corresponds a stochastically-continuous process with stationary independent increments
 +
 
 +
$$
 +
X  =  \{ {X ( t) } : {0 \leq  t < \infty } \}
 +
,\  X ( 0)  = 0 ,
 +
$$
  
 
such that
 
such that
  
<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/l058/l058270/l05827040.png" /></td> </tr></table>
+
$$
 +
F ( X)  = {\mathsf P} \{ X ( 1) < x \} .
 +
$$
  
In turn, a [[Separable process|separable process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827041.png" /> of the type mentioned has with probability 1 sample trajectories without discontinuities of the second kind; hence for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827042.png" /> the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827043.png" /> equal to the number of elements in the set
+
In turn, a [[Separable process|separable process]] $  X $
 +
of the type mentioned has with probability 1 sample trajectories without discontinuities of the second kind; hence for $  b > a > 0 $
 +
the random variable $  Y ( [ a , b ) ) $
 +
equal to the number of elements in the set
  
<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/l058/l058270/l05827044.png" /></td> </tr></table>
+
$$
 +
\left \{ {t } : {a \leq  \lim\limits _ {\tau \downarrow 0 } \
 +
X ( t + \tau ) - \lim\limits _ {\tau \downarrow 0 } \
 +
X ( t - \tau ) < b , 0 \leq  t \leq  1 } \right \}
 +
,
 +
$$
  
i.e. to the number of jumps with heights in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827045.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827046.png" />, exists. In this notation, one has for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827047.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827048.png" />,
+
i.e. to the number of jumps with heights in $  [ a , b ) $
 +
on the interval $  [ 0 , 1 ] $,
 +
exists. In this notation, one has for the function $  N $
 +
corresponding to $  F $,
  
<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/l058/l058270/l05827049.png" /></td> </tr></table>
+
$$
 +
{\mathsf E} \{ Y ( [ a , b ) ) \}  = N ( b) - N ( a) .
 +
$$
  
A similar relation holds for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827050.png" />.
+
A similar relation holds for the function $  M $.
  
Many properties of the behaviour of the sample trajectories of a separable process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827051.png" /> can be expressed in terms of the characteristics of the Lévy canonical representation of the distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827052.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827053.png" />,
+
Many properties of the behaviour of the sample trajectories of a separable process $  X $
 +
can be expressed in terms of the characteristics of the Lévy canonical representation of the distribution function $  {\mathsf P} \{ X ( 1) < x \} $.  
 +
In particular, if $  \sigma  ^ {2} = 0 $,
  
<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/l058/l058270/l05827054.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {x \rightarrow 0 }  N ( x)  > - \infty ,\ \
 +
\lim\limits _ {x \rightarrow 0 }  M ( x)  < \infty ,
 +
$$
  
<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/l058/l058270/l05827055.png" /></td> </tr></table>
+
$$
 +
\gamma  = \int\limits _ {- \infty } ^ { 0 } 
 +
\frac{x}{1 + x  ^ {2} }
 +
  d M
 +
( x) + \int\limits _ { 0 } ^  \infty 
 +
\frac{x}{1 + x  ^ {2} }
 +
  d N ( x) ,
 +
$$
  
then almost-all the sample functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827056.png" /> are with probability 1 step functions with finitely many jumps on any finite interval. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827057.png" /> and if
+
then almost-all the sample functions of $  X $
 +
are with probability 1 step functions with finitely many jumps on any finite interval. If $  \sigma  ^ {2} = 0 $
 +
and if
  
<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/l058/l058270/l05827058.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { - 1} ^ { 0 }  | x |  d M ( x) +
 +
\int\limits _ { 0 } ^ { 1 }  x  d N ( x)  < \infty ,
 +
$$
  
then with probability 1 the sample trajectories of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827059.png" /> have bounded variation on any finite interval. Directly in terms of the characteristics of the Lévy canonical representation one can calculated the [[Infinitesimal operator|infinitesimal operator]] corresponding to the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058270/l05827060.png" />, regarded as a Markov random function. Many analytical properties of an infinitely-divisible distribution function can be expressed directly in terms of the characteristics of its Lévy canonical representation.
+
then with probability 1 the sample trajectories of $  X $
 +
have bounded variation on any finite interval. Directly in terms of the characteristics of the Lévy canonical representation one can calculated the [[Infinitesimal operator|infinitesimal operator]] corresponding to the process $  X $,  
 +
regarded as a Markov random function. Many analytical properties of an infinitely-divisible distribution function can be expressed directly in terms of the characteristics of its Lévy canonical representation.
  
 
There are analogues of the Lévy canonical representation for infinitely-divisible distributions given on a wide class of algebraic structures.
 
There are analogues of the Lévy canonical representation for infinitely-divisible distributions given on a wide class of algebraic structures.
  
 
====References====
 
====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"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><TR><TD valign="top">[4]</TD> <TD valign="top">  I.I. [I.I. Gikhman] Gihman,   A.V. [A.V. Skorokhod] Skorohod,   "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K. Itô,   "Stochastic processes" , Aarhus Univ. (1969)</TD></TR></table>
+
{|
 
+
|valign="top"|{{Ref|GK}}|| B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) {{MR|0062975}} {{ZBL|0056.36001}}
 
+
|-
 +
|valign="top"|{{Ref|Pe}}|| V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) {{MR|0388499}} {{ZBL|0322.60043}} {{ZBL|0322.60042}}
 +
|-
 +
|valign="top"|{{Ref|PR}}|| Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) {{MR|0251754}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|GS}}|| I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , '''2''' , Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}}
 +
|-
 +
|valign="top"|{{Ref|I}}|| K. Itô, "Stochastic processes" , Aarhus Univ. (1969)
 +
|}
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Loève,   "Probability theory" , '''1''' , Springer (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.P. Breiman,   "Probability" , Addison-Wesley (1968)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"E. Lukacs,   "Characteristic functions" , Griffin (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Heyer,   "Probability measures on locally compact groups" , Springer (1977)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K.R. Parthasarathy,   "Probability measures on metric spaces" , Acad. Press (1967)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"B.V. Gnedenko,   A.N. Kolmogorov,   "Introduction to the theory of random processes" , Saunders (1969) (Translated from Russian)</TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|Lo}}|| M. Loève, "Probability theory" , '''1''' , Springer (1977) {{MR|0651017}} {{MR|0651018}} {{ZBL|0359.60001}}
 +
|-
 +
|valign="top"|{{Ref|B}}|| L.P. Breiman, "Probability" , Addison-Wesley (1968) {{MR|0229267}} {{ZBL|0174.48801}}
 +
|-
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|valign="top"|{{Ref|Lu}}|| E. Lukacs, "Characteristic functions" , Griffin (1970) {{MR|0346874}} {{MR|0259980}} {{ZBL|0201.20404}} {{ZBL|0198.23804}}
 +
|-
 +
|valign="top"|{{Ref|H}}|| H. Heyer, "Probability measures on locally compact groups" , Springer (1977)
 +
|-
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|valign="top"|{{Ref|Pa}}|| K.R. Parthasarathy, "Probability measures on metric spaces" , Acad. Press (1967) {{MR|0226684}} {{ZBL|0153.19101}}
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|-
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|valign="top"|{{Ref|GK2}}|| B.V. Gnedenko, A.N. Kolmogorov, "Introduction to the theory of random processes" , Saunders (1969) (Translated from Russian)
 +
|}

Latest revision as of 01:14, 19 January 2022


2020 Mathematics Subject Classification: Primary: 60E07 Secondary: 60G51 [MSN][ZBL]

A formula for the logarithm $ \mathop{\rm ln} \phi ( \lambda ) $ of the characteristic function of an infinitely-divisible distribution:

$$ \mathop{\rm ln} \phi ( \lambda ) = i \gamma \lambda - \frac{\sigma ^ {2} \lambda ^ {2} }{2} + \int\limits _ {- \infty } ^ { 0 } \left ( e ^ {i \lambda x } - 1 - \frac{i \lambda x }{1 + x ^ {2} } \right ) \ d M ( x) + $$

$$ + \int\limits _ { 0 } ^ \infty \left ( e ^ {i \lambda x } - 1 - \frac{i \lambda x }{1 + x ^ {2} } \right ) d N ( x) , $$

where the characteristics of the Lévy canonical representation, $ \gamma $, $ \sigma ^ {2} $, $ M $, and $ N $, satisfy the following conditions: $ - \infty < \gamma < \infty $, $ \sigma ^ {2} \geq 0 $, and $ M ( x) $ and $ N ( x) $ are non-decreasing left-continuous functions on $ ( - \infty , 0 ) $ and $ ( 0 , \infty ) $, respectively, such that

$$ \lim\limits _ {x \rightarrow \infty } \ N ( x) = \lim\limits _ {x \rightarrow - \infty } \ M ( x) = 0 $$

and

$$ \int\limits _ { - 1} ^ { 0 } x ^ {2} d M ( x) < \infty ,\ \ \int\limits _ { 0 } ^ { 1 } x ^ {2} d N ( x) < \infty . $$

To every infinitely-divisible distribution there corresponds a unique system of characteristics $ \gamma $, $ \sigma ^ {2} $, $ M $, $ N $ in the Lévy canonical representation, and conversely, under the above conditions on $ \gamma $, $ \sigma ^ {2} $, $ M $, and $ N $ the Lévy canonical representation with respect to such a system determines the logarithm of the characteristic function of some infinitely-divisible distribution.

Thus, for the normal distribution with mean $ a $ and variance $ \sigma ^ {2} $:

$$ \gamma = a ,\ \sigma ^ {2} = \sigma ^ {2} ,\ \ N ( x) \equiv 0 ,\ M ( x) \equiv 0 . $$

For the Poisson distribution with parameter $ \lambda $:

$$ \gamma = \frac \lambda {2} ,\ \ \sigma ^ {2} = 0 ,\ \ M ( x) \equiv 0 ,\ \ N ( x) = \left \{ \begin{array}{rl} - \lambda & \textrm{ for } x \leq 1 , \\ 0 & \textrm{ for } x > 1 . \\ \end{array} \right .$$

To the stable distribution with exponent $ \alpha $, $ 0 < \alpha < 2 $, corresponds the Lévy representation with

$$ \sigma ^ {2} = 0 ,\ \ \textrm{ any } \ \gamma ,\ M ( x) = \frac{c _ {1} }{| x | ^ \alpha } ,\ \ N ( x) = - \frac{c _ {2} }{x ^ \alpha } , $$

where $ c _ {i} \geq 0 $, $ i = 1 , 2 $, are constants $ ( c _ {1} + c _ {2} > 0 ) $. The Lévy canonical representation of an infinitely-divisible distribution was proposed by P. Lévy in 1934. It is a generalization of a formula found by A.N. Kolmogorov in 1932 for the case when the infinitely-divisible distribution has finite variance. For $ \mathop{\rm ln} \phi ( \lambda ) $ there is a formula equivalent to the Lévy canonical representation, proposed in 1937 by A.Ya. Khinchin and called the Lévy–Khinchin canonical representation. The probabilistic meaning of the functions $ N $ and $ M $ and the range of applicability of the Lévy canonical representation are defined as follows: To every infinitely-divisible distribution function $ F $ corresponds a stochastically-continuous process with stationary independent increments

$$ X = \{ {X ( t) } : {0 \leq t < \infty } \} ,\ X ( 0) = 0 , $$

such that

$$ F ( X) = {\mathsf P} \{ X ( 1) < x \} . $$

In turn, a separable process $ X $ of the type mentioned has with probability 1 sample trajectories without discontinuities of the second kind; hence for $ b > a > 0 $ the random variable $ Y ( [ a , b ) ) $ equal to the number of elements in the set

$$ \left \{ {t } : {a \leq \lim\limits _ {\tau \downarrow 0 } \ X ( t + \tau ) - \lim\limits _ {\tau \downarrow 0 } \ X ( t - \tau ) < b , 0 \leq t \leq 1 } \right \} , $$

i.e. to the number of jumps with heights in $ [ a , b ) $ on the interval $ [ 0 , 1 ] $, exists. In this notation, one has for the function $ N $ corresponding to $ F $,

$$ {\mathsf E} \{ Y ( [ a , b ) ) \} = N ( b) - N ( a) . $$

A similar relation holds for the function $ M $.

Many properties of the behaviour of the sample trajectories of a separable process $ X $ can be expressed in terms of the characteristics of the Lévy canonical representation of the distribution function $ {\mathsf P} \{ X ( 1) < x \} $. In particular, if $ \sigma ^ {2} = 0 $,

$$ \lim\limits _ {x \rightarrow 0 } N ( x) > - \infty ,\ \ \lim\limits _ {x \rightarrow 0 } M ( x) < \infty , $$

$$ \gamma = \int\limits _ {- \infty } ^ { 0 } \frac{x}{1 + x ^ {2} } d M ( x) + \int\limits _ { 0 } ^ \infty \frac{x}{1 + x ^ {2} } d N ( x) , $$

then almost-all the sample functions of $ X $ are with probability 1 step functions with finitely many jumps on any finite interval. If $ \sigma ^ {2} = 0 $ and if

$$ \int\limits _ { - 1} ^ { 0 } | x | d M ( x) + \int\limits _ { 0 } ^ { 1 } x d N ( x) < \infty , $$

then with probability 1 the sample trajectories of $ X $ have bounded variation on any finite interval. Directly in terms of the characteristics of the Lévy canonical representation one can calculated the infinitesimal operator corresponding to the process $ X $, regarded as a Markov random function. Many analytical properties of an infinitely-divisible distribution function can be expressed directly in terms of the characteristics of its Lévy canonical representation.

There are analogues of the Lévy canonical representation for infinitely-divisible distributions given on a wide class of algebraic structures.

References

[GK] B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) MR0062975 Zbl 0056.36001
[Pe] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) MR0388499 Zbl 0322.60043 Zbl 0322.60042
[PR] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) MR0251754
[GS] I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027
[I] K. Itô, "Stochastic processes" , Aarhus Univ. (1969)

Comments

References

[Lo] M. Loève, "Probability theory" , 1 , Springer (1977) MR0651017 MR0651018 Zbl 0359.60001
[B] L.P. Breiman, "Probability" , Addison-Wesley (1968) MR0229267 Zbl 0174.48801
[Lu] E. Lukacs, "Characteristic functions" , Griffin (1970) MR0346874 MR0259980 Zbl 0201.20404 Zbl 0198.23804
[H] H. Heyer, "Probability measures on locally compact groups" , Springer (1977)
[Pa] K.R. Parthasarathy, "Probability measures on metric spaces" , Acad. Press (1967) MR0226684 Zbl 0153.19101
[GK2] B.V. Gnedenko, A.N. Kolmogorov, "Introduction to the theory of random processes" , Saunders (1969) (Translated from Russian)
How to Cite This Entry:
Lévy canonical representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%A9vy_canonical_representation&oldid=22727
This article was adapted from an original article by B.A. Rogozin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article