Abstract
A new class of multidimensional locally perturbed random walks called random walks with sticky barriers is introduced and analyzed. The laws of large numbers and functional limit theorems are proved for hitting times of successive barriers.
Highlights
A standard random walk formed by partial sums of independent identically distributed random vectors is one of the most simple, classical and well-studied discretetime random processes
The first model of this flavor is due to Harrison and Shepp [5] who investigated simple random walk on Z with a perturbation at state 0
If the process is at state x ∈ Z \ {0} it jumps to x ± 1 with probabilities 1/2, whereas from the state 0 it jumps to ±1 with probability p ∈ [0, 1] \ {1/2}
Summary
A standard random walk formed by partial sums of independent identically distributed random vectors is one of the most simple, classical and well-studied discretetime random processes. Recent results and generalizations of this model can be found in [7, 9] and [10] Another representative of the class of locally perturbed random walks is a planar random walk in a semi-infinite strip studied in [2]. Motivated by the aforementioned study we introduce and analyze in this note a new class of multidimensional locally perturbed random walks, which we call random walks with sticky barriers (RWSB) and which generalizes the model treated in [2]. One can drop the assumption that the increments are Nd0 -valued and define a RWSB with state space Rd. Most of the results obtained in this paper hold in this more general settings upon necessary amendments.
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