necessary condition for anomalous diffusion. We note also that in this case one may expect that the rescaled probability density P ( ψ, t ) at t → ∞ approaches the stable probability density, as the generalized central limit theorem [18] suggests. The coefficient of proportionality a and the exponent α are in general not independent and can be found from the asymptotic behavior of the probability density u ( g ) in the vicinity of the point g = − g$_{0}$ . Indeed, using the condition ∫ g$_{0}$ − g$_{0}$ dg gu ( g ) = 0, which is a consequence of the symmetry property of u ( g ), from Eq. (2.9) at f = g$_{0}$ we obtain ( τ → ∞ , κ → ∞ ). Then, assuming that where g → − g$_{0}$ , b > 0 and β > 0, the asymptotic formula (3.3) takes the form Finally, comparing (3.5) with (3.2), we find α = β and where a$_{od}$ = bg$_{0}$ ( νl/g$_{0}$ ) α is the parameter a in the overdamped limit. According to these results, the particle mass decreases the parameter a in comparison with the overdamped case but does not change the exponent α . An example of u ( g ) having the asymptotic behavior (3.4) is the probability density where Γ( x ) = ∫ ∞ 0 dy y x − $^{1}$e − y is the gamma function, which corresponds to the symmetric beta distribution. According to Eq. (3.7), this distribution is unimodal with the maximum at g = 0 if β > 1, bimodal with infinite maxima at g = ± g$_{0}$ if β < 1, and uniform if β = 1. Determining the parameter b directly from the density function (3.7), for this example we obtain p ( τ ) ∼ νl τ 2 [ 1 − g$_{0}$ κνl ( 1 − τ ∂ ∂τ )] u ( νl τ − g$_{0}$ ) (3.3) u ( g ) ∼ b ( 1 + g g$_{0}$ ) β − 1 , (3.4) p ( τ ) ∼ bg$_{0}$ ( νl g$_{0}$ ) $^{β}$( 1 − βg$_{0}$ κνl ) 1 τ 1+ β . (3.5) a = a$_{od}$ ( 1 − αg$_{0}$ κνl ) , (3.6) u ( g ) = Γ( β + 1 / 2) $_{g$_{0}$}$√$_{π}$$_{Γ(}$$_{β}$$_{)}$ ( 1 − g 2 g 2 $_{0}$) β − 1 , (3.7) a$_{od}$ = Γ( α + 1 / 2) $_{2}$√$_{π}$$_{Γ(}$$_{α}$$_{)}$ ( 2 νl g$_{0}$ ) α . (3.8) 8