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In this paper, we propose a new interpretation of local limit theorems for univariate and multivariate distributions on lattices. We show that - given a local limit theorem in the standard sense - the distributions are approximated well by the limit distribution, uniformly on intervals of possibly decaying length. We identify the maximally allowable decay speed of the interval lengths. Further, we show that for continuous distributions, the interval type local law holds without any decay speed restrictions on the interval lengths. We show that various examples fit within this framework, such as standardized sums of i.i.d. random vectors or correlated random vectors induced by multidimensional spin models from statistical mechanics.
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We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that the process survives for a large time t, building on previous results by Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, providing an essentially complete answer to a question first addressed by Kesten (1978). An important tool in the proofs of these results is the convergence of a certain observable to a continuous state branching process. Our proofs incorporate new ideas which might be of use in other branching models.
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 establish several optimal moment comparison inequalities (Khinchin-type inequalities) for weighted sums of independent identically distributed symmetric discrete random variables which are uniform on sets of consecutive integers. Specifically, we obtain sharp constants for the second moment and any moment of order at least 3 (using convex dominance by Gaussian random variables). In the case of only 3 atoms, we also establish a Schur-convexity result. For moments of order less than 2, we get sharp constants in two cases by exploiting Haagerups arguments for random signs.
We explore an asymptotic behavior of densities of sums of independent random variables that are convoluted with a small continuous noise.
Let $r=r(n)$ be a sequence of integers such that $rleq n$ and let $X_1,ldots,X_{r+1}$ be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on $mathbb{R}^n$. Limit theorems for the log-volume and the volume of the random convex hull of $X_1,ldots,X_{r+1}$ are established in high dimensions, that is, as $r$ and $n$ tend to infinity simultaneously. This includes, Berry-Esseen-type central limit theorems, log-normal limit theorems, moderate and large deviations. Also different types of mod-$phi$ convergence are derived. The results heavily depend on the asymptotic growth of $r$ relative to $n$. For example, we prove that the fluctuations of the volume of the simplex are normal (respectively, log-normal) if $r=o(n)$ (respectively, $rsim alpha n$ for some $0 < alpha < 1$).
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