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Let ζ be a Markov chain on a finite state space D, f a function from D to ℝd , and S n = ∑k=1 n f(ζk ). We prove an invariance theorem for S and derive an explicit expression of the limit covariance matrix. We give its exact value for p-reinforced random walks on ℤ2 with p = 1, 2, 3.

Correlated random walk models figure prominently in many scientific disciplines. Of fundamental importance in such applications is the development of the characteristic function of the n-step probability distribution since it contains complete information on the probability structure of the process. Using a simple algebraic lemma we derive the n-step characteristic function of the Gillis correlated random walk together with other related results. In particular, we present a new and simple proof of Gillis's conjecture, consider the generalization to the Gillis–Domb–Fisher walk, and examine the effect of including an arbitrary initial distribution.

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Un principe d'invariance pour une classe de marches p-corrélées sur ℤd

The Gillis–Domb–Fisher correlated random walk

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2 results

Un principe d'invariance pour une classe de marches p-corrélées sur ℤd

The Gillis–Domb–Fisher correlated random walk