Gambler’s ruin with random stopping

Abstract

. Let { X j , j ≥ 0 } denote a Markov process on [ − N − 1 , N +1] ∪{ c } . Suppose P ( X j +1 = m +1 | X j = m ) = ph , P ( X j +1 = m − 1 | X j = m ) = (1 − p ) h , all j ≥ 1 and | m | ≤ N , where p = 12 + bN and h = 1 − c N for c N = 12 a 2 /N 2 . Define P ( X j +1 = c | X j = m ) = c N , j ≥ 0, | m | ≤ N . { X j } terminates at the first j such that X j ∈ {− N − 1 , N + 1 , c } . Let L = max { j ≥ 0 : X j = 0 } . On Ω ◦ = { X j terminates at c } , denote by R ◦ , V ◦ , and L ◦ respectively, as the numbers of runs, short runs, and steps from L until termination. Denote Y ◦ = R ◦ − 2 V ◦ and Z ◦ = L ◦ − 3 R ◦ +2 V ◦ . Then lim N →∞ E { e i 1 N ( s Y ◦ + t Z ◦ ) | Ω ◦ } = C a,b