Lévy canonical representation

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)

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)
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[GK2]	B.V. Gnedenko, A.N. Kolmogorov, "Introduction to the theory of random processes" , Saunders (1969) (Translated from Russian)