Fractional Edgeworth expansions for one-dimensional heavy-tailed random variables and applications. (English) Zbl 07733577

Chiarini, Leandro; Jara, Milton; Ruszel, Wioletta M.

Electron. J. Probab. 28, Paper No. 108, 42 p. (2023).

The authors study a class of lattice random variables in the domain of attraction of an \(\alpha\)-stable random variable with index \(\alpha \in (0,2)\) which satisfy a truncated fractional Edgeworth expansion.
The behaviour of the class under simple operations is considered, with specific examples. Also, sharp rates of convergence to an \(\alpha\)-stable distribution in a local central limit theorem are examined as well as Green’s function expansions. The authors also explore fluctuations of a class of discrete stochastic PDE’s driven by the heavy-tailed random walks belonging to the class of fractional Edgeworth expansions.

Reviewer: Pavel Stoynov (Sofia)

MSC:

60J45	Probabilistic potential theory
60G50	Sums of independent random variables; random walks
60E10	Characteristic functions; other transforms
60G52	Stable stochastic processes
60E07	Infinitely divisible distributions; stable distributions
60F05	Central limit and other weak theorems

Keywords:

discrete stochastic linear stochastic equations; fluctuations; fractional Edgeworth expansion; heavy-tailed random walks; local central limit theorem; potential kernel; stable distributions

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