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Let S=(Sk)k≥0 be a random walk on ℤ and ξ=(ξi)i∈ℤ a stationary random sequence of centered random variables, independent of S. We consider a random walk in random scenery that is the sequence of random variables (Un)n≥0, where Un=∑k=0nξSk, n∈ℕ. Under a weak dependence assumption on the scenery ξ we prove a functional limit theorem generalizing Kesten and Spitzer’s [Z. Wahrsch. Verw. Gebiete50 (1979) 5–25] theorem.
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