Let ${X_i}$ be a process with stationary and independent increments whose log characteristic function is expressed as $ibut-2^{-1}sigma^2u^2t+tint_{{0 }^c}{(exp(iux)-1-iux(i+x^2)^{-1})dv(x)}$. Our main result is taht $x^2(int_{y>x}{dv(y)})/(int_{$mid$y$mid$leqx}{y^2dv(y)+sigma^2}) o 1$</TEX> as $x o 0 (resp. x o infty)$ is necessary, and sufficient for ${X-i}$ to have ${A_t}$ and ${B_t}$ such that $(X_t-A_t)/B_t o^D n(0,1)$ as $t o 0 (resp. t o infty)$.