Random Walk in Random and Non-Random Environments

Shao's theorem on the maximum of standardized random walk increments for multidimensional arrays

“…+ X n , S 0 = 0 be the random walk generated by {X i } and let 2) be the maximum of standardized random walk increments. In his book [14] Révész gave upper and lower bounds for the almost-sure limiting behavior of L n in the case of Bernoulli-distributed summands [14], Theorem 14. 16, and conjectured that in this case lim n→∞ L n / √ 2 log n = 1 a.s. Establishing a general version of Révész conjecture, Shao [16] proved that lim n→∞ L n / 2 log n = α * ∈ [1, ∞] a.s., (1.3) Keywords and phrases.…”

Maximal Local Time of a d-dimensional Simple Random Walk on Subsets

“…Taking into account the well-known fact that log ρ n ∼ 2 log n (see e.g. [9], p. 115) a simple application of our Theorem 1.1 implies that for the maximal local time of the Cauchy walk we have 1 2π ≤ lim inf n→∞ η(n) (log n) 2 ≤ lim sup n→∞ η(n) (log n) 2 ≤ 2 π a.s.…”

“…For d = 1 the interested reader should consult the monograph of P. Révész [9]. In this paper we are interested in investigating the maximum local time for d ≥ 2 in a restricted sense, namely we want to investigate the maximum on certain subsets of the state space.…”

The fractional mixed fractional brownian motion and fractional brownian sheet

“…-Last but not least, this paper extends El-Nouty's results [7][8][9][10][11] and consequently answers some new questions. Recall first two definitions of the Lévy classes, stated in Révész [22]. Let {Z(t), t ≥ 0} be a stochastic process defined on the basic probability space (Ω, A).…”