The Asymptotic Behaviour Of One-Dimensional Nearest-Neighbour Random Walks In Random Environments

Suppose that random environment q = {q(n)}₀^∞, is i.i.d. and valued on [0,1] ,RWRE {S_n}₀^∞, S₀ = 0, V integers x_t{i≥ 0), x,y, satisfies0, others.it's first-arrive times are defined as, at 0 : T₀ = 0, at integer n≠0: T_n = ini{k : S_k = n}, it's first-arrive time difference series: Ï„_n = T_n-T_n-1, (n > 0). Ï„_-n, T_-n. is defined similarly. We get the distribution of T_n, for positive integer n,the expectation of Ï„_,Ï„_-1further, the series {Ï„_n}₁^∞ is i.i.d..Following all the foregoing, we find out it's asymptotic behavior : and prove it's L.L.N. :(i)p < 1/2 impliesafter averaged with respect to the law of the random environment.