Sankhya: The Indian Journal of Statistics

1998, Volume 60, Series A, Pt. 2, pp. 151-170

A NOTE ON A DISTRIBUTION OF WEIGHTED SUMS OF I.I.D. RAYLEIGH RANDOM VARIABLES

By

P. HITCZENKO, *North Carolina State University, Raleigh*

*SUMMARY.* In this note we show that if $X,X_1,X_2,\dots$
are
independent and identically distributed random variables with density $f_X(t)=
2t\exp(-t^2)$, $t>0$ and $(a_1,\dots, a_n)$ are nonnegative numbers such that
$\sum_{k=1}^na_k^2=1$, then for every $t>0$ the following inequality is true
$$P(\sum_{k=1}^na_kX_k\ge t)\le P({1\over\sqrt n}\sum_{k=1}^nX_k\ge t).$$ We
will actually prove a comparison inequality in a more general context.

*AMS (1991) subject classification.* 60E15, 60G50.

*Key words and phrases. * tail probability,
weighted sum, Rayleigh distribution.