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{{MSC|60E07|60G51}}

[[Category:Distribution theory]]
[[Category:Stochastic processes]]

A formula for the logarithm  $   \mathop{\rm ln}  \phi ( \lambda ) $
of the [[Characteristic function|characteristic function]] of an [[Infinitely-divisible distribution|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|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|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|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|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|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|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====
{|
|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====

====References====
{|
|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}}
|-
|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)
|-
|valign="top"|{{Ref|Pa}}|| K.R. Parthasarathy, "Probability measures on metric spaces" , Acad. Press (1967) {{MR|0226684}} {{ZBL|0153.19101}}
|-
|valign="top"|{{Ref|GK2}}|| B.V. Gnedenko, A.N. Kolmogorov, "Introduction to the theory of random processes" , Saunders (1969) (Translated from Russian)
|}