Green Function for an Asymptotically Stable Random Walk in a Half Space

• Published in: Journal of theoretical probability Vol. 37; no. 2; pp. 1745 - 1786
• Main Authors: Denisov, Denis, Wachtel, Vitali
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2024; Springer Nature B.V
• Subjects: Asymptotic properties; Green's functions; Infinity; Mathematics; Mathematics and Statistics; Probability Theory and Stochastic Processes; Random walk; Statistics; Secondary 60G40; Random walk; Conditioned process; Exit time; Primary 60G50; 60F17; Harmonic function; 60J45
• ISSN: 0894-9840; 1572-9230
• DOI: 10.1007/s10959-023-01283-4

We consider an asymptotically stable multidimensional random walk S ( n ) = ( S 1 ( n ) , … , S d ( n ) ) . For every vector x = ( x 1 … , x d ) with x 1 ≥ 0 , let τ x : = min { n > 0 : x 1 + S 1 ( n ) ≤ 0 } be the first time the random walk x + S ( n ) leaves the upper half space. We obtain the asymptotics of p n ( x , y ) : = P ( x + S ( n ) ∈ y + Δ , τ x > n ) as n tends to infinity, where Δ is a fixed cube. From that, we obtain the local asymptotics for the Green function G ( x , y ) : = ∑ n p n ( x , y ) , as | y | and/or | x | tend to infinity.