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

✍ Scribed by Bernd Jesiak

• Publisher: John Wiley and Sons
• Year: 1980
• Tongue: English
• Weight: 343 KB
• Volume: 95
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19800950116

✦ Synopsis

Limit theorems under the assumption of restricted convergence on a set with a finite limit point Von BERND JESIAP, Leipdg (Eingegangen am 27. 10. 1978)

Summary. We consider double sequences { ( X n k ) } of independent and asymptotically constant kn random variables. For certain constants A , we put F,(x) = P 2 Xnk -A,-=x . The central

problem is the following one: Assume that {F,} for x E I c R, converges weakly to a noncomtant limit function YJ where I is a set with a finite limit point restricted convergence F, -Y . Then additional conditions ensuring the complete convergence F, * 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 . I n the case of identically distributed summands with common distribution function V the assumption ) (

s sufficient to prove F,=. Fa, provided that F , -. F , ( F a stands for a stable distribution with characteristic exponent a).