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Limit theorems for one and two-dimensional random walks in random scenery

Fabienne CastellNadine Guillotin-PlantardFrançoise Pène — 2013

Annales de l'I.H.P. Probabilités et statistiques

Random walks in random scenery are processes defined by Z n : = k = 1 n ξ X 1 + + X k , where ( X k , k 1 ) and ( ξ y , y d ) are two independent sequences of i.i.d. random variables with values in d and respectively. We suppose that the distributions of X 1 and ξ 0 belong to the normal basin of attraction of stable distribution of index α ( 0 , 2 ] and β ( 0 , 2 ] . When d = 1 and α 1 , a functional limit theorem has been established in ( (1979) 5–25) and a local limit theorem in (To appear). In this paper, we establish the convergence in distribution and a local...

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