**Sankhya: The Indian Journal of Statistics**

1996, Volume 58, Series A, Pt. 1, pp. 1--7

**COMPLEX VALUED CHARACTERISTIC FUNCTIONS WITH (SOME) REAL POWERS**

By

B. RAMACHANDRAN, *Indian Statistical Institute*

SUMMARY. It is shown that, for every positive integer $n \geq 2$, there exist distribution functions F on the Real line such that (i) the convolutions F^{*r} are asymmetric for $1 \leq r \lt n$, while F^{*n} is symmetric; and (ii) F has absolute moments of all orders < 11 -providing a partial answer to a question raised by Staudte and Tata. (In two sequels, it is shown that (ii) it can be replaced, respectively, by : (ii') F has absolute moments upto and including / not including any order p 0; (ii'') F has moments of all orders.)

*AMS (1990) subject classification.* Primary 60E10; secondary 60E05.

*Key words and phrases.* Characteristic function, symmetric asymmetric distributions, moments, infinite divisibility.