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On normal characterizations by the distribution of linear forms, assuming finite variance

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  • Arnold, Barry C.
  • Isaacson, Dean L.

Abstract

If X1 and X2 are independent and identically distributed (i. i. d.) with finite variance, then (X1+X2)/[radical sign]2 has the same distribution as X1 if and only if X1 is normal with mean zero (Pólya [9]). The idea of using linear combinations of i. i. d. random variables to characterize the normal has since been extended to the case where [sigma][infinity]i=1aiXi has the same distribution as X1. In particular if at least two of the ai's are non-zero and X1 has finite variance, then Laha and Lukacs [8] showed that X1 is normal. They also [7] established the same result without the assumption of finite variance. The purpose of this note is to present a different and easier proof of the characterization under the assumption of finite variance. The idea of the proof follows closely the approach used by Pólya in [9]. The same technique is also used to give a characterization of the exponential distribution.

Suggested Citation

  • Arnold, Barry C. & Isaacson, Dean L., 1978. "On normal characterizations by the distribution of linear forms, assuming finite variance," Stochastic Processes and their Applications, Elsevier, vol. 7(2), pages 227-230, June.
    • Handle: RePEc:eee:spapps:v:7:y:1978:i:2:p:227-230
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    Cited by:

    1. Yanushkevichius, Romanas, 2000. "Stability of the characterization of normal distribution in the Laha-Lukacs theorem," Statistics & Probability Letters, Elsevier, vol. 49(3), pages 225-233, September.

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