2018/02/27 by Kôhei Uchiyama, Uchiyama, Kohei · 2 citations
Decision Sciences · Mathematics · #60G50 #60J45 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1802.09832
openalex publication_date 2018/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Sn =X1+⋯ +Xn be an irreducible random walk (r.w.) on the one dimensional integer lattice with zero mean, infinite variance and i.i.d. increments Xn. We obtain an upper and lower bounds of the potential function, a(x), of Sn in the form a(x)\asymp x/m(x) under a reasonable condition on the distribution of Xn; we especially show that as x→∞ a(x) \asymp (x)/(m-(x)) \quadand (a(-x))/(a(x)) → 0 if limx→ +∞ (m+(x))/(m-(x)) =0, where m_±(x) = ∫0xdy∫y^∞ P[± X1>u]du and m=m++m-. Under certain conditions on the tails of the distribution of X we derive precise asymptotic forms of a(x) as x→ +∞ or/and -∞. The results are applied to derive a sufficient condition for the relative stability of the ladder height and estimates of some escape probabilities from the origin; we show among others that under the above condition on m+/m-, P[Sn>0] → 1/α if and only if the probability of exiting a long interval [-Q,R] through the upper boundary converges to λα-1 as Q/(Q+R) → λ for any 0