content_image/341956.jpg content_image/341957.jpg content_image/341958.jpg content_image/341959.jpg
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. 

## E. α = 2 

In this case we use the following representation for the Laplace transform of p ( τ ): 

<img src='content_image/341956.jpg'>

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 

<img src='content_image/341957.jpg'>

<img src='content_image/341958.jpg'>

Using this result, the Laplace transforms of the first two moments of Y$_{t}$ at s → 0 can be written as 

<img src='content_image/341959.jpg'>

