It should be noted that similar asymptotic formulas for 〈 Y$_{t}$ 〉 and 〈 Y 2 t 〉 were obtained in Ref. [20]. But because of the use of the waiting time probability density p ( τ ) of a particular form, the asymptotic formulas derived in that paper do not depend on the parameter a . At the same time, as it was shown above, the parameter a contains an important information about the role of quenched disorder and particle mass. In particular, Eq. (3.6) shows that the moments (4.28) increase with the particle mass. Since lim$_{t}$ →∞ 〈 Y$_{t}$ 〉 $^{2}$/ 〈 Y 2 t 〉 = 1, for finding σ $^{2}$( t ) as t → ∞ we need to know at least the two leading terms of the long-time expansion of 〈 Y$_{t}$ 〉 and 〈 Y 2 t 〉 . In principle, using Eq. (4.25) and the integral term in Eq. (4.24), we could easily find the asymptotic behavior of 1 − p$_{s}$ − as ln(1 /s ) as s → 0 and, in this way, obtain the second terms of the asymptotic expansion of 〈 Y$_{t}$ 〉 s and 〈 Y 2 t 〉 $_{s}$. However, in contrast to the previous case, a straightforward application of the Tauberian theorem to this case does not provide a precise determination of the second terms of the asymptotic expansion of 〈 Y$_{t}$ 〉 and 〈 Y 2 t 〉 . As it was argued in Sec. IV A, in order to find these terms it is necessary to go beyond the Tauberian theorem. In this case we use the following representation for the Laplace transform of p ( τ ): E. α = 2 where This form of p$_{s}$ explicitly accounts for both the finiteness of τ and the asymptotic behavior of p ( τ ). Since p ( τ ) − a/τ 3 = o (1 /τ $^{3}$) ( τ → ∞ ), at s → 0 we can neglect the integral term in Eq. (4.29) in comparison with ar$_{s}$ . This, together with the asymptotic formula r$_{s}$ ∼ − ( s $^{2}$/ 2) ln(1 /s ) ( s → 0), yields Using this result, the Laplace transforms of the first two moments of Y$_{t}$ at s → 0 can be written as p$_{s}$ = 1 − τs − ar$_{s}$ − ∫ ∞ τ$_{min}$ dτ (1 − sτ − e − $^{sτ}$) ( p ( τ ) − a τ 3 ) , (4.29) r$_{s}$ = ∫ ∞ τ$_{min}$ dτ 1 τ $_{3}$(1 − sτ − e − $^{sτ}$) = s $_{2}$1 − 2 ξ − (1 − ξ ) e − ξ 2 ξ 2 − s 2 2 E$_{1}$ ( ξ ) . (4.30) p$_{s}$ − 1 + τ s ∼ a 2 s $^{2}$ln 1 s . (4.31) 〈 Y$_{t}$ 〉 s − l τ 1 s 2 ∼ la 2 τ 2 1 s ln 1 s (4.32) 14