Refined behaviour of a conditioned random walk in the large deviations regime

TY - JOUR

T1 - Refined behaviour of a conditioned random walk in the large deviations regime

AU - Asmussen, Søren

AU - Glynn, Peter W.

N1 - Publisher Copyright: © 2024 ISI/BS.

PY - 2024/2

Y1 - 2024/2

N2 - Conditioned limit theorems as n →∞ are given for the increments X1, …, Xn of a random walk Sn = X1 +· · · + Xn, subject to the conditionings Sn ≥ nb or Sn = nb with b > EX. The probabilities of these conditioning events are given by saddlepoint approximations, corresponding to the exponential tilting (formula presented) of the increment density (formula presented). It has been noted in various formulations that conditionally, the increment density somehow is close to fθ (x). Sharp versions of such statements are given, including correction terms for segments (X1, …, Xk) with k fixed. Similar correction terms are given for the mean and variance of (formula presented) where (formula presented) is the empirical c.d.f. of X1, …, Xn. Also a result on the (total variation) distance for segments with k/n → c ∈ (0, 1) is derived. Further functional limit theorems for ̂(formula presented) are given, involving a bivariate conditioned Brownian limit.

AB - Conditioned limit theorems as n →∞ are given for the increments X1, …, Xn of a random walk Sn = X1 +· · · + Xn, subject to the conditionings Sn ≥ nb or Sn = nb with b > EX. The probabilities of these conditioning events are given by saddlepoint approximations, corresponding to the exponential tilting (formula presented) of the increment density (formula presented). It has been noted in various formulations that conditionally, the increment density somehow is close to fθ (x). Sharp versions of such statements are given, including correction terms for segments (X1, …, Xk) with k fixed. Similar correction terms are given for the mean and variance of (formula presented) where (formula presented) is the empirical c.d.f. of X1, …, Xn. Also a result on the (total variation) distance for segments with k/n → c ∈ (0, 1) is derived. Further functional limit theorems for ̂(formula presented) are given, involving a bivariate conditioned Brownian limit.

KW - Boltzmann law

KW - conditioned Brownian motion

KW - empirical c.d.f

KW - exponential tilting

KW - functional limit theorem

KW - Gibbs conditioning

KW - saddlepoint approximation

KW - total variation distance

UR - http://www.scopus.com/inward/record.url?scp=85177228283&partnerID=8YFLogxK

U2 - 10.3150/23-BEJ1601

DO - 10.3150/23-BEJ1601

M3 - Journal article

AN - SCOPUS:85177228283

SN - 1350-7265

VL - 30

SP - 371

EP - 387

JO - Bernoulli

JF - Bernoulli

IS - 1

ER -