# Lévy canonical representation

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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)