Large Deviation Principles for Random Walk Trajectories. I

Abstract

This paper deals with a random walk $S_n:=\xi_1+\cdots+\xi_n$, $n=0,1,\ldots,$ in the $d$-dimensional Euclidean space ${\mathbb R}^d$, where $S_0=0$ and $\xi_k$ are independent identically distributed random vectors satisfying Cramér's moment conditions. For random polygons with nodes at the points $(\frac{k}{n},\frac{1}{x}S_k)$, $k=0,1,\ldots,n,$ we obtain the logarithmic asymptotics of the large deviation probabilities in different trajectory spaces when $x\sim \alpha_0 n$, $\alpha_0>0$, as $n\to\infty.$ The results include the so-called local and extended large deviation principles (l.d.p.'s) (see i̧te15) that hold in those cases where the “usual” l.d.p. does not apply. The paper consists of three parts. Part I has two sections. Section 1 presents the key concepts and some facts concerning the l.d.p. in arbitrary metric spaces. In section 2 we formulate the “strong” versions of the “usual” l.d.p. in the large deviation zones that were obtained earlier in [A. A. Borovkov, Theory Probab. Appl., 12 (1967), pp. 575--595], [A. A. Mogul'skii, Theory Probab. Appl., 21 (1976), pp. 300--315] for the space of continuous functions. Besides that, section 2 also contains the l.d.p. for probabilities for the random walk trajectories to hit a convex set. That result was obtained using inequalities from [A. A. Borovkov and A. A. Mogul'skii, Theory Probab. Appl., 56 (2012), pp. 21--43] and does not involve any moment conditions. Part II begins with section 3 presenting an example elucidating the need to extend both the problem formulation and the very concept of the “large deviation principle.” We introduce a new extended functional space, a metric therein, and the deviation functional (integral) of a more general kind that will be used when constructing an “extended” l.d.p. In section 4 we present and prove the key results of the paper, the local and the extended large deviation principles, for the trajectories of univariate random walks in the space ${\mathbb D}$ of functions without discontinuities of the second kind. Section 5 extends to the multivariate case all the results established in section 4. Section 6 in Part III presents results analogous to those from section 4, but now established in the space of functions of bounded variation with a metric stronger than that in $\mathbb D$. In section 7 we establish the so-called conditional large deviation principles for the trajectories of univariate random walks given the location of the walk at the terminal point. As a consequence, we obtain the Sanov's theorem on large deviations of empirical distributions.