Conditioned local limit theorems for random walks on the real line

2021/10/11 by Grama, Ion, Xiao, Hui
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2110.05123

Abstract

Consider a random walk S_n=∑_i=1ⁿ X_i with independent and identically distributed real-valued increments X_i of zero mean and finite variance. Assume that X_i is non-lattice and has a moment of order 2+δ. For any x≥ 0, let τ_x = inf \ k≥ 1: x+S_k < 0 \ be the first time when the random walk x+S_n leaves the half-line [0,∞). We study the asymptotic behavior of the probability \bb P (τ_x >n) and that of the expectation 𝔼 ( f(x + S_n ), τ_x > n ) for a large class of target function f and various values of x, y possibly depending on n. This general setting implies limit theorems for the joint distribution ℙ ( x + S_n ∈ y+ [0, Δ], τ_x > n ) where Δ>0 may also depend on n. In particular, the case of moderate deviations y=σ√(q nlog n) is considered. We also deduce some new asymptotics for random walks with drift and give explicit constants in the asymptotic of the probability \bb P (τ_x =n). For the proofs we establish new conditioned integral limit theorems with precise error terms.

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