Let G be a probability distribution on \([0,\infty)\) with the properties: 1) \(G([x,\infty))>0\) for every \(x>0,\) 2) \(G^{2*}([x,\infty))/G([x,\infty))\to 2\int_{R}e^{\gamma u}dG(u)=\hat G(\gamma)\) as \(x\to \infty\), for some \(\gamma\geq 0,\) where * denotes convolution, 3) for every real number y, \[ G([x+y,\infty))/G([x,\infty))\to e^{- \gamma y}\text{ as } x\to \infty. \] The main result of the paper is the following theorem: Let F be an infinitely divisible distribution on R with Lévy's spectral measure \(\nu\) and let G be a probability distribution with the properties 1)-3).Then the following assertions are equivalent: a) \(F([x,\infty))\sim c_ 1G([x,\infty))\) for some \(c_ 1>0\) as \(x\to \infty,\) b) \(\nu([x,\infty))\sim c_ 2G([x,\infty))\) for some \(c_ 2>0\) as \(x\to \infty,\) c) F satisfies the condition 3), \(\hat F(\gamma)<\infty\) and \(F([x,\infty))\sim \hat F(\gamma)\nu ([x,\infty))\) as \(x\to \infty.\)