Local probabilities for random walks conditioned to stay positive

2007/11/08 by Vladimir Vatutin, Vatutin, Vladimir, Vitali Wachtel +1 · 2 citations
Decision Sciences · Mathematics · Economics, Econometrics and Finance · #Probability and Risk Models #Stochastic processes and statistical mechanics #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.0711.1302

Abstract

Let S₀=0,S_n, n>0 be a random walk generated by a sequence of i.i.d. random variables X₁,X₂,... and let τ^- be the first descending ladder epoch. Assuming that the distribution of X₁ belongs to the domain of attraction of an α-stable law we study the asymptotic behavior of the local probabilities P(τ^-=n) and the conditional local probabilities P(S_n∈ [x,x+y)|τ^->n) for fixed y and x=x(n)∈ (0,∞).

Cited by

• Limit theorems for critical branching processes in an extremely unfavorable random environment
• Reduced critical branching processes in non-favorable random environment

Related

• Local limit theorems for ladder epochs
• Local behaviour of first passage probabilities
• Expansions for random walks conditioned to stay positive
• Random walks conditioned to stay non-negative and branching processes in non-favorable random environment
• Invariance principles for local times at the supremum of random walks and Lévy processes