Vol. 15, No. 1, 1965

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On a linear form whose distribution is identical with that of a monomial

Radha Govinda Laha and Eugene Lukacs

Vol. 15 (1965), No. 1, 207–214

Several authors studied identically distributed linear forms in independently and identically distributed random variables. J. Marcinkiewicz considered finite or infinite linear forms and assumed that the random variables have finite moments of all orders. He showed that the common distribution of the random variables is then the Normal distribution. Yu. V. Linnik obtained some deep results concerning identically distributed linear forms involving only a finite number of random variables. The authors have investigated in a separate paper the case where one of the linear forms contains infinitely many terms while the other is a monomial. They obtained a characterization of the normal distribution under the assumption that the second moment of the random variable is finite. In the present paper we investigate a similar problem and do not assume the existence of the second moment.

Mathematical Subject Classification
Primary: 60.20
Received: 14 October 1963
Revised: 10 March 1964
Published: 1 March 1965
Radha Govinda Laha
Eugene Lukacs