Return Probabilities for the Reflected Random Walk on $$\mathbb{N }_0$$ N 0

Rim Essifi () and Marc Peigné ()
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Rim Essifi: LMPT, UMR 7350
Marc Peigné: LMPT, UMR 7350

Journal of Theoretical Probability, 2015, vol. 28, issue 1, 231-258

Abstract: Abstract Let $$(Y_n)$$ ( Y n ) be a sequence of i.i.d. $$\mathbb{Z }$$ Z -valued random variables with law $$\mu $$ μ . The reflected random walk $$(X_n)$$ ( X n ) is defined recursively by $$X_0=x \in \mathbb{N }_0, X_{n+1}=\vert X_n+Y_{n+1}\vert $$ X 0 = x ∈ N 0 , X n + 1 = | X n + Y n + 1 | . Under mild hypotheses on the law $$\mu $$ μ , it is proved that, for any $$ y \in \mathbb{N }_0$$ y ∈ N 0 , as $$n \rightarrow +\infty $$ n → + ∞ , one gets $$\mathbb{P }_x[X_n=y]\sim C_{x, y} R^{-n} n^{-3/2}$$ P x [ X n = y ] ∼ C x , y R − n n − 3 / 2 when $$\sum _{k\in \mathbb{Z }} k\mu (k) >0$$ ∑ k ∈ Z k μ ( k ) > 0 and $$\mathbb{P }_x[X_n=y]\sim C_{ y} n^{-1/2}$$ P x [ X n = y ] ∼ C y n − 1 / 2 when $$\sum _{k\in \mathbb{Z }} k\mu (k) =0$$ ∑ k ∈ Z k μ ( k ) = 0 , for some constants $$R, C_{x, y}$$ R , C x , y and $$C_y >0$$ C y > 0 .

Keywords: Random walks; Local limit theorem; Generating function; Wiener-Hopf factorization; 60J10; 60J15; 60B15; 60F15 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10959-013-0490-3

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