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

We consider double sequences {{X_nk}} of independent and asymptotically constant random variables. For certain constants A_n we put The central problem is the following one: Assume that {F_n} for x ? I ? R₁ converges weakly to a nonconstant limit function ψ where I is a set with a finite limit point (restricted convergence Then additional conditions ensuring the complete convergence F_n ? F to a certain limit distribution function F are 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 function V the assumption a sufficient to prove F_n ? F_α, provided that (F_α stands for a stable distribution with characteristic exponent α).