THE LAW OF LARGE NUMBERS FOR RANDOM WALKS IN RANDOM SCENERY WITH UNCORRELATED TERMS

In this paper, we prove the law of large numbers for a random walk in random scenery. The limiting behavior of such sequences has been intensively studied since the 1980s. Such results, in particular, allow proving the consistency of statistical estimates of unknown parameters in many situations. Unlike previous results, we allow the terms of the random walk, on whose states the random walk in a random scenery is built, to have different distributions and not be centered. We also do not require that the terms of the random walk in a random scenery be identically distributed and independent; it is only required that they have the same mean and be uncorrelated. The research methods are classical methods of probability theory: various probabilistic inequalities (Berry–Esseen, H¨older’s, Lyapunov’s), as well as limit theorems (the central limit theorem, the law of large numbers). It should be noted that the model under consideration has a physical interpretation associated with the motion of a particle in a random environment.

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