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

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 fθ(x)= eθx−ψ(θ)f(x) of the increment density f(x), with θ satisfying b= EθX=ψ′(θ) where ψ(θ)=logEeθX. 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 Fˆn(x)−Fθ(x) where Fˆn 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 (Fˆk(x),Sk)k≤n are given, involving a bivariate conditioned Brownian limit.