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We consider a Levy process on the line that is transient and with nonpolar one point sets. For a > 0 let A^a) be the total occupation time of [0, a] and R(a) the Lebesgue measure of the range of the process intersected with [0, a]. Whenever [0, oo) is a recurrent set we show N(a)/EN(a) -R(a)/ER(a) converges in the mean square to 0 as a -> oo. This in turn is used to derive limit laws for R(a)/ER(a) from those for N(a)/EN(a).doi:10.1090/s0002-9939-1988-0955017-x fatcat:23pumbpf4vehljp3pkpcosjxou