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LARGE DEVIATION PRINCIPLES FOR RANDOM WALK TRAJECTORIES. I?

机译：随机游走轨迹的大偏差原理。一世？

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This paper deals with a random walk S_n: = ξ_1 + ··· + ξ_n, n = 0, 1,..., in the ddimensional Euclidean space Rd, where S_0 = 0 and ξ_k are independent identically distributed random vectors satisfying Cramér's moment conditions. For random polygons with nodes at the points (k/ n, 1/xS_k), k = 0, 1,..., n, we obtain the logarithmic asymptotics of the large deviation probabilities in different trajectory spaces when x ～ α_0n, α_0 > 0, as n→∞. The results include the so-called local and extended large deviation principles (l.d.p.'s) (see [1]) 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 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 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.
机译：本文处理一个随机游动S_n：=ξ_1+··+ +ξ_n，n = 0，1，...，在三维欧几里德空间Rd中，其中S_0 = 0和ξ_k是满足Cramér矩的独立均匀分布的随机矢量条件。对于节点为（k / n，1 / xS_k），k = 0，1，...，n的随机多边形，当x〜α_0n，α_0时，我们获得了不同轨迹空间中大偏差概率的对数渐近性> 0，如n→∞。结果包括所谓的局部和扩展大偏差原理（l.d.p。）（请参阅[1]），在那些情况下，“通常” l.d.p.不适用。本文由三部分组成。第一部分有两个部分。第1节介绍了有关l.d.p.的关键概念和一些事实。在任意度量空间中。在第2节中，我们制定了“通常” l.d.p.的“强”版本。在[A．博罗夫科夫（A. Borovkov），Problem Probab。 Appl。，12（1967），pp。575-595]，[A. A. Mogul'skii，理论专家。 Appl。，21（1976），pp。300-315]。除此之外，第2节还包含l.d.p。随机行走轨迹碰到凸集的概率。该结果是使用[A. A. Borovkov和A. A. Mogul'skii，理论专家。 Appl。，56（2012），pp。21-43]，并且不涉及任何时刻条件。第二部分从第3节开始，给出一个示例，阐明需要扩展问题表述和“大偏差原理”的概念。我们介绍了一个新的扩展功能空间，一个度量，以及一种更通用的偏差函数（积分），它在构建“扩展” l.d.p.时将使用。在第4节中，我们介绍并证明了本文的主要结果，局部和扩展的大偏差原理，用于函数空间D中单变量随机游动的轨迹，而没有第二种不连续性。第5节将所有在第4节中确定的结果扩展到多变量情况。第三部分第6节介绍的结果类似于第4节中的结果，但是现在在有界变异函数的空间中建立了度量，该度量的强于D中的度量。在第7节中，我们给出了在终点位置的单变量随机游动轨迹的所谓条件大偏差原理。结果，我们获得了经验分布大偏差的Sanov定理。

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