Conditioned limit theorems as are given for the increments of a random walk , subject to the conditionings or with . The probabilities of these conditioning events are given by saddlepoint approximations, corresponding to the exponential tilting of the increment density , with θ satisfying where . It has been noted in various formulations that conditionally, the increment density somehow is close to . Sharp versions of such statements are given, including correction terms for segments with k fixed. Similar correction terms are given for the mean and variance of where is the empirical c.d.f. of . Also a result on the total variation distance for segments with is derived. Further functional limit theorems for are given, involving a bivariate conditioned Brownian limit.
Bernoulli
30(1):
371-387
(February 2024).
DOI: 10.3150/23-BEJ1601
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