CHARACTERIZATIONS OF THE POWER FUNCTION DISTRIBUTION BY THE INDEPENDENCE OF THE LOWER RECORD VALUES

This paper presents characterizations of the power distribution with the parameter $\beta=1$ by the independence of the lower record values. We prove $X\;{\in}\;POW({\alpha},\;1)$ for ${\alpha}\;>\;0$, if and only if $\frac{X_{L(n)}}{X_{L(m)}}$ and $X_{L(m)}$ for $1\;{\leq}\;m\;<\;n$ are independent. And we prove that $X\;{\in}\;POW({\alpha},\;1)$ for ${\alpha}\;>\;0$, if and only if $\frac{X_{L(m)}-X_{L(m+1)}}{X_{L(m)}}$ and $X_{L(m)$ for $m\;{\geq}\;1$ are independent or $\frac{X_{L(m)}-X_{L(m+1)}}{X_{L(m+1)}}$ and $X_{L(m)}$ for $m\;{\geq}\;1$ are independent.