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

Abstract

Conditioned limit theorems as $n\to \mathrm{\infty }$ are given for the increments $X_1,\dots ,X_n$ of a random walk $S_n=X_1+\cdots +X_n$, subject to the conditionings $S_n\ge nb$ or $S_n=nb$ with $b>\mathbb{E}X$. The probabilities of these conditioning events are given by saddlepoint approximations, corresponding to the exponential tilting $f_{\mathit{\theta }}(x)=$ ${\mathrm{e}}^{\mathit{\theta }x-\mathit{\psi }(\mathit{\theta })}f(x)$ of the increment density $f(x)$, with θ satisfying $b={\mathbb{E}}_{\mathit{\theta }}X={\mathit{\psi }}^{\prime }(\mathit{\theta })$ where $\mathit{\psi }(\mathit{\theta })=log\mathbb{E}{\mathrm{e}}^{\mathit{\theta }X}$. It has been noted in various formulations that conditionally, the increment density somehow is close to $f_{\mathit{\theta }}(x)$. Sharp versions of such statements are given, including correction terms for segments $(X_1,\dots ,X_k)$ with k fixed. Similar correction terms are given for the mean and variance of ${\hat{F}}_n(x)-F_{\mathit{\theta }}(x)$ where ${\hat{F}}_n$ is the empirical c.d.f. of $X_1,\dots ,X_n$. Also a result on the total variation distance for segments with $k∕n\to c\in (0,1)$ is derived. Further functional limit theorems for ${({\hat{F}}_k(x),S_k)}_{k\le n}$ are given, involving a bivariate conditioned Brownian limit.