Domains of Semi-Stable Attraction of Nonnormal Semi-Stable Laws

✍ Scribed by H.P. Scheffler

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 387 KB
• Volume: 51
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1994.1071

✦ Synopsis

A sequence of independent, identically distributed random vectors (X_{1}, X_{2}, \ldots) is said to belong to the (Q)-normed domain of semi-stable attraction of a random vector (Y) if there exist diagonal matrices (A_{n}), constant vectors (b_{n}) and a sequence (\left(k_{n}\right){n}) of natural numbers with (k{n} \uparrow \infty) and (k_{n+1} / k_{n} \rightarrow c \geqslant 1) such that (A_{n}\left(X_{1}+\cdots+X_{k_{n}}\right)+b_{n}) converges in distribution to (Y). The limit law (Y) is then called semi-stable. We present a simple, necessary, and sufficient condition for the existence of such (A_{n}, b_{n}), and (k_{n}) in the case where (Y) has no normal component. Furthermore we prove some moment conditions for random vectors belonging to the (Q)-normed domain of semi-stable attraction of (Y, 1994) Academic Press, Inc.