Central limit theorem for sampled sums of dependent random variables

Abstract

Abstract. We prove a central limit theorem for linear triangular arrays under weak dependence conditions. Our result is then applied to dependent random variables sampled by a Z-valued transient random walk. This extends the results obtained by [N. Guillotin-Plantard and D. Schneider, Stoch. Dynamics 3 (2003) 477-497]. An application to parametric estimation by random sampling is also provided.

Empirical processes for recurrent and transient random walks in random scenery

“…Far from being exhaustive, we can cite strong approximation results and laws of the iterated logarithm [15,16,31], limit theorems for correlated sceneries or walks [26,27,14], large and moderate deviations results [1,22,10,11], ergodic and mixing properties (see the survey [20]).…”

Quenched central limit theorems for random walks in random scenery

“…we prove that for P-almost every fixed path of the random scenery, a limit theorem holds for Z n (correctly renormalized). It is worth remarking that when the random walk S is fixed, functional limit theorems for the sequence (Z [nt] ) t≥0 have been proved (see [17,23,21]). Indeed, conditionally to the random walk, the sum Z n can be viewed as a sum of IID random variables weighted by the local time of the random walk.…”

Asymptotic variance of the self-intersections of stable random walks using Darboux-Wiener theory

“…In this direction we now give the following result on the asymptotic moments of the self-intersection local time V n , For α ∈ Z, let N α (n) = n j=0 1 S j =α . The proof of a.s. convergence of V n /EV n → 1 is essentially given in [12] but relies heavily on the bound var(V n ) = O(n 2 ). The rest be easily adapted from [1].…”