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

Seminar in Insurance and Economics

SPEAKER: Søren Asmussen (Aarhus University).

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

ABSTRACT: Conditioned limit theorems as $n\to\infty$ are given for the increments $X_1,\ldots,X_n$ 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 $f_\theta(x)=$ $e^{\theta x-\psi(\theta)}f(x)$ of the increment density $f(x)$, with $\theta$ satisfying $b=\mathbb{E}_\theta X=\psi'(\theta)$ where $\psi(\theta)=\log\mathbb{E} e^{\theta X}$. It has been noted in various formulations that conditionally, the increment density somehow is close to $f_\theta(x)$, with Martin-Löf's Boltzmann law as an early example. Sharp versions of such statements are given, including correction terms for segments $(X_1,\ldots,X_k)$ with $k$ fixed. Similar correction terms are given for the mean and variance of $\widehat F_n(x)-F_\theta(x)$ where $\widehat F_n$ is the empirical c.ds.f. of $X_1,\ldots,X_n$. 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 $(\widehat F_k(x),S_k)_{k\le n}$ are given, involving a bivariate conditioned Brownian limit.

Joint work with Peter Glynn (Stanford), to appear in Bernoulli.

Link to Seminar in Insurance and Economics.