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Register a new userPrecise large deviations of sums of widely dependent random variables and its applications
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In this paper, we obtain some results on precise large deviations for non-random and random sums of widely dependent random variables with common dominatedly varying tail distribution or consistently varying tail distribution on $(-infty,infty)$. Then we apply the results to reinsurance and insurance and give some asymptotic estimates on proportional reinsurance, random-time ruin probability and the finite-time ruin probability.
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We consider the problem of bounding large deviations for non-i.i.d. random variables that are allowed to have arbitrary dependencies. Previous works typically assumed a specific dependence structure, namely the existence of independent components. Bounds that depend on the degree of dependence between the observations have only been studied in the theory of mixing processes, where variables are time-ordered. Here, we introduce a new way of measuring dependences within an unordered set of variables. We prove concentration inequalities, that apply to any set of random variables, but benefit from the presence of weak dependencies. We also discuss applications and extensions of our results to related problems of machine learning and large deviations.
Let ${{bf mathcal{Z}}_n:ngeq 1}$ be a sequence of i.i.d. random probability measures. Independently, for each $ngeq 1$, let $(X_{n1},ldots, X_{nn})$ be a random vector of positive random variables that add up to one. This paper studies the large deviation principles for the randomly weighted sum $sum_{i=1}^{n} X_{ni} mathcal{Z}_i$. In the case of finite Dirichlet weighted sum of Dirac measures, we obtain an explicit form for the rate function. It provides a new measurement of divergence between probabilities. As applications, we obtain the large deviation principles for a class of randomly weighted means including the Dirichlet mean and the corresponding posterior mean. We also identify the minima of relative entropy with mean constraint in both forward and reverse directions.
We study one-dimensional nearest neighbour random walk in site-random environment. We establish precise (sharp) large deviations in the so-called ballistic regime, when the random walk drifts to the right with linear speed. In the sub-ballistic regime, when the speed is sublinear, we describe the precise probability of slowdown.
Let {(X_i,Y_i)}_{i=1}^n be a sequence of independent bivariate random vectors. In this paper, we establish a refined Cramer type moderate deviation theorem for the general self-normalized sum sum_{i=1}^n X_i/(sum_{i=1}^n Y_i^2)^{1/2}, which unifies and extends the classical Cramer (1938) theorem and the self-normalized Cramer type moderate deviation theorems by Jing, Shao and Wang (2003) as well as the further refined version by Wang (2011). The advantage of our result is evidenced through successful applications to weakly dependent random variables and self-normalized winsorized mean. Specifically, by applying our new framework on general self-normalized sum, we significantly improve Cramer type moderate deviation theorems for one-dependent random variables, geometrically beta-mixing random variables and causal processes under geometrical moment contraction. As an additional application, we also derive the Cramer type moderate deviation theorems for self-normalized winsorized mean.
We prove a Large Deviations Principle for the number of intersections of two independent infinite-time ranges in dimension five and more, improving upon the moment bounds of Khanin, Mazel, Shlosman and Sina{i} [KMSS94]. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen and den Hollander [BBH04], who analyzed this question for the Wiener sausage in finite-time horizon. The proof builds on their result (which was resumed in the discrete setting by Phetpradap [Phet12]), and combines it with a series of tools that were developed in recent works of the authors [AS17, AS19a, AS20]. Moreover, we show that most of the intersection occurs in a single box where both walks realize an occupation density of order one.
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