totic formula (3.2), we obtain 1 − p$_{s}$ ∼ s $^{α}$a ∫ ∞ 0 dx 1 − e − x x 1+ α (4.6) as s → 0. An integration by parts together with the integral representation of the gamma function [21], Γ( x ) = ∫ ∞ 0 dy e − $^{y}$yx − $^{1}$, reduces (4.6) to the form 1 − p$_{s}$ ∼ a Γ(1 − α ) α s $^{α}$. (4.7) Now, using this result and the Laplace transforms 〈 Y$_{t}$ 〉 s = l p$_{s}$ s (1 − p$_{s}$ ) , 〈 Y 2 t 〉 s = l 2 p 2 s + p$_{s}$ s (1 − p$_{s}$ ) 2 (4.8) of the first two moments of Y$_{t}$ , we find in the limit s → 0: 〈 Y$_{t}$ 〉 s ∼ lα a Γ(1 − α ) 1 s 1+ α (4.9) and 〈 Y 2 t 〉 s ∼ 2 l $^{2}$α2 a $^{2}$Γ$^{2}$(1 − α ) 1 s 1+2 $_{α}$. (4.10) Since these asymptotic formulas are particular cases of the asymptotic formula (4.1) in which the slowly varying function L (1 /s ) is a constant, from (4.2) we obtain in the long-time limit 〈 Y$_{t}$ 〉 ∼ lα a Γ(1 − α )Γ(1 + α ) t α (4.11) and 〈 Y 2 t 〉 ∼ 2 l $^{2}$α2 a $^{2}$Γ$^{2}$(1 − α )Γ(1 + 2 α ) t 2 $^{α}$. (4.12) Thus, in this case lim$_{t}$ →∞ 〈 Y$_{t}$ 〉 $^{2}$/ 〈 Y 2 t 〉 ̸ = 1 and the above asymptotic expressions yield σ $^{2}$( t ) ∼ l $^{2}$α2 a $^{2}$Γ$^{2}$(1 − α ) ( 2 Γ(1 + 2 α ) − 1 Γ $^{2}$(1 + α ) ) t 2 $^{α}$. (4.13) According to this result, which agrees with that obtained in the context of the asymptotic solution of the CTRW [20], subdiffusion occurs if α ∈ (0 , 1 / 2) and superdiffusion if α ∈ (1 / 2 , 1). If α = 1 / 2 then σ $^{2}$( t ) ∝ t and, in accordance with the commonly used terminology, the biased diffusion is normal. However, for normal diffusion processes both the mean and variance are proportional to time. Therefore, since 〈 Y$_{t}$ 〉 ∝ t 1 / 2 at α = 1 / 2, this type of diffusion should be more appropriately termed as quasi-normal. It is also worthy to note that, according to Eqs. (3.6) and (4.13), the larger is the particle mass, the stronger is diffusion. 11