Error estimates of approximation by stable laws. I. (English. Russian original) Zbl 0923.60037

Nagaev, S. V.

Theory Probab. Math. Stat. 56, 151-165 (1998); translation from Teor. Jmovirn. Mat. Stat. 56, 145-160 (1997).

The problem of approximation by a stable law may be formulated as follows: for a given distribution function \(F\) and a given \(n\) find a stable law \(V_{n}(x)\) such that \[ \sup_{x}| F^{*n}(x)-V_{n}(x)| =\inf_{\{V\}}\sup_{x}| F^{*n}-V(x)|, \] where \(\{V\}\) is the set of all stable laws. The author deals with the following distance: \[ \Delta_{n}:=\sup_{x}| F^{*n}(x)-V^{*n}(x)| , \] where \(V\) is an arbitrary stable law. Estimate of error \(\Delta_{n}\) of approximation by a stable law is given.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

60F10	Large deviations
60E05	Probability distributions: general theory
60E07	Infinitely divisible distributions; stable distributions

Keywords:

stable law; estimate of error; large deviations