Stability of the characterization of normal distribution in the Laha–Lukacs theorem

✍ Scribed by Romanas Yanushkevichius

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 109 KB
• Volume: 49
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(00)00051-1

✦ Synopsis

If X1 and X2 are independent and identically distributed (i.i.d.) random variables with ÿnite variance, then (X1 + X2)= √ 2 has the same distribution as X1 if and only if X1 is normal with mean zero (Polya, 1923, Math. Zeitschrift 18, 96 -108). About ten authors devoted their works to stability problems of this characterization. The idea of using linear combinations of i.i.d. random variables to characterize the normal distribution has been extended by Laha and Lukacs (1965) to the case where ∞ i=1 aiXi has the same distribution as X1. We investigate the stability of this characterization.