Limit theorems under the assumption of restricted convergence on a set with a finite limit point

摘要

AbstractWe consider double sequences {{Xnk}} of independent and asymptotically constant random variables. For certain constantsAnwe put The central problem is the following one: Assume that {Fn} forxϵI⊂R1converges weakly to a nonconstant limit function ψ whereIis a set with a finite limit point (restricted convergence Then additional conditions ensuring the complete convergenceFn⟹Fto a certain limit distribution functionFare given. These additional conditions are weaker than the corresponding sufficient conditions known from the classical theory. Further, these results yield two new versions of the central limit theorem, see § 2. In the case of identically distributed summands with common distribution functionVthe assumption a sufficient to proveFn⟹Fα, provided that (Fαstands for a stable distribution with characteristic e