Finally, applying to Eq. (2.12) the inverse Laplace transform defined as h ( t ) = L − $^{1}${ h$_{s}$ } = (1 / 2 πi $^{)}$∫ c + i ∞ c − i ∞ ds e $^{st}$h$_{s}$ ( c is chosen to be larger than the real parts of all singularities of h$_{s}$ ), we find the first and the second As it follows from the waiting time probability density (2.9), the m th moment of the waiting time, τ m = ∫ ∞ 0 dτ τ $^{m}$p ( τ ) ( m = 1 , 2 , . . . ), can be written in the form ′ m Since the probability density u ( g ) is normalized on the interval [ − g$_{0}$, g$_{0}$ ], from Eq. (3.1) it follows that τ m ≤ ( νl + 2 g$_{0}$/κ ) $^{m}$/ ( f − g$_{0}$ ) $^{m}$. Thus, if f > g$_{0}$ then all these moments are finite, and so in this case the classical central limit theorem for sums of a random number of random variables [17] is applied to Y$_{t}$ . This implies that σ $^{2}$( t ) ∝ t as t → ∞ , i.e., at f > g$_{0}$ the biased diffusion of particles is normal, and the rescaled probability density P ( ψ, t ) = σ ( t ) P ( 〈 Y$_{t}$ 〉 + σ ( t ) ψ, t ) in the long-time limit tends to the probability density P ( ψ, ∞ ) = (2 π ) − 1 / $^{2}$e − ψ $^{2}$/ 2 of the standard normal distribution. It is clear from the above that the anomalous long-time behavior of the variance σ $^{2}$( t ) is expected at τ 2 = ∞ when the mentioned central limit theorem becomes inapplicable. The condition τ 2 = ∞ implies f = g$_{0}$ that, in accordance with (2.4), yields τ$_{min}$ = νl/ 2 g$_{0}$ and τ$_{max}$ = ∞ . Since the divergence of τ 2 occurs when p ( τ ) at τ → ∞ tends to zero slowly enough, next we assume that ( τ → ∞ ) with a > 0 and α ∈ (0 , 2]. Thus, the biased diffusion in a randomly layered medium is expected to be anomalous if both conditions, f = g$_{0}$ and α ∈ (0 , 2], hold. The former guarantees that the waiting time τ can be arbitrarily large ( τ$_{max}$ = ∞ ), and so it is a 〈 Y$_{t}$ 〉 = l L − 1 { p$_{s}$ s (1 − p$_{s}$ ) } (2.13) σ $^{2}$( t ) = 〈 Y 2 t 〉 − 〈 Y$_{t}$ 〉 $^{2}$. (2.15) moments of Y$_{t}$ , which in turn determine the variance of the particle position: 〈 Y 2 t 〉 = l $^{2}$L − 1 { p 2 s + p$_{s}$ s (1 − p$_{s}$ ) 2 } (2.14) τ m = ∫ g$_{0}$ − g$_{0}$ ∫ g$_{0}$ − g$_{0}$ dgdg $^{′}$u ( g ) u ( g $^{′}$) ( κνl + g − g ′ κ ( f + g ) ) . (3.1) p ( τ ) ∼ a τ 1+ α (3.2) 7 III. CONDITIONS OF ANOMALOUS DIFFUSION