Lévy canonical representation

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

A formula for the logarithm of the characteristic function of an infinitely-divisible distribution:

where the characteristics of the Lévy canonical representation, , , , and , satisfy the following conditions: , , and and are non-decreasing left-continuous functions on and , respectively, such that

and

To every infinitely-divisible distribution there corresponds a unique system of characteristics , , , in the Lévy canonical representation, and conversely, under the above conditions on , , , and 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 and variance :

For the Poisson distribution with parameter :

To the stable distribution with exponent , , corresponds the Lévy representation with

where , , are constants . 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 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 and and the range of applicability of the Lévy canonical representation are defined as follows: To every infinitely-divisible distribution function corresponds a stochastically-continuous process with stationary independent increments

such that

In turn, a separable process of the type mentioned has with probability 1 sample trajectories without discontinuities of the second kind; hence for the random variable equal to the number of elements in the set

i.e. to the number of jumps with heights in on the interval , exists. In this notation, one has for the function corresponding to ,

A similar relation holds for the function .

Many properties of the behaviour of the sample trajectories of a separable process can be expressed in terms of the characteristics of the Lévy canonical representation of the distribution function . In particular, if ,

then almost-all the sample functions of are with probability 1 step functions with finitely many jumps on any finite interval. If and if

then with probability 1 the sample trajectories of 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 , 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

[1]	B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian) MR0062975 Zbl 0056.36001
[2]	V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian) MR0388499 Zbl 0322.60043 Zbl 0322.60042
[3]	Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian) MR0251754
[4]	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
[5]	K. Itô, "Stochastic processes" , Aarhus Univ. (1969)

References

[a1]	M. Loève, "Probability theory" , 1 , Springer (1977) MR0651017 MR0651018 Zbl 0359.60001
[a2]	L.P. Breiman, "Probability" , Addison-Wesley (1968) MR0229267 Zbl 0174.48801
[a3]	E. Lukacs, "Characteristic functions" , Griffin (1970) MR0346874 MR0259980 Zbl 0201.20404 Zbl 0198.23804
[a4]	H. Heyer, "Probability measures on locally compact groups" , Springer (1977)
[a5]	K.R. Parthasarathy, "Probability measures on metric spaces" , Acad. Press (1967) MR0226684 Zbl 0153.19101
[a6]	B.V. Gnedenko, A.N. Kolmogorov, "Introduction to the theory of random processes" , Saunders (1969) (Translated from Russian)