A probabilistic approach to the tails of infinitely divisible laws.

*(English)*Zbl 0728.60019
Sums, Trimmed sums and extremes, Prog. Probab. 23, 317-335 (1991).

[For the entire collection see Zbl 0717.00017.]

The Lévy-Khinchine canonical representation for an infinitely divisible characteristic function is used to represent the corresponding random variable as the sum of three independent random variables, one with a normal distribution, the other two basically compound Poisson. Using this representation, and properties of the Poisson distribution, the authors give proofs of several results on the tail behaviour of inf div distributions, such as given by V. M. Kruglov [Theory Probab. Appl. 15(1970), 319-324 (1971); translation from Teor. Veroyatn. Primen. 15, 330-336 (1970; Zbl 0301.60014)], K. Sato [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 12, 101-109 (1973; Zbl 0279.60010)] and many others.

The authors’ point of view is interesting, and they obtain some new results on one-sided tails. Their claim, however, that their “entire theory becomes independent of the existing literature”, seems a bit far- fetched. They start from the canonical representation, without which Sato’s results cannot even be formulated, and from there kind of rediscover the connection with Lévy processes, such as discussed in e.g. Breiman’s probability. The proofs are analytic rather than probabilistic; they use properties of the distributions of the representing random variables, which are, in fact, the canonical measures.

The Lévy-Khinchine canonical representation for an infinitely divisible characteristic function is used to represent the corresponding random variable as the sum of three independent random variables, one with a normal distribution, the other two basically compound Poisson. Using this representation, and properties of the Poisson distribution, the authors give proofs of several results on the tail behaviour of inf div distributions, such as given by V. M. Kruglov [Theory Probab. Appl. 15(1970), 319-324 (1971); translation from Teor. Veroyatn. Primen. 15, 330-336 (1970; Zbl 0301.60014)], K. Sato [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 12, 101-109 (1973; Zbl 0279.60010)] and many others.

The authors’ point of view is interesting, and they obtain some new results on one-sided tails. Their claim, however, that their “entire theory becomes independent of the existing literature”, seems a bit far- fetched. They start from the canonical representation, without which Sato’s results cannot even be formulated, and from there kind of rediscover the connection with Lévy processes, such as discussed in e.g. Breiman’s probability. The proofs are analytic rather than probabilistic; they use properties of the distributions of the representing random variables, which are, in fact, the canonical measures.

Reviewer: F.W.Steutel (Eindhoven)

##### MSC:

60E07 | Infinitely divisible distributions; stable distributions |

60E15 | Inequalities; stochastic orderings |