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Large deviations for uniform projections of p-radial distributions on lnp - balls

Kaufmann Tom, Sambale Holger, Thäle Christoph

Probability and Mathematical Statistics

2022

Vol. 42, Fasc. 2

303--317

We consider products of uniform random variables from the Stiefel manifold of orthonormal k-frames in Rn, k ≤ n, and random vectors from the n-dimensional ℓnp-ball Bnp with certain p-radial distributions, p ∈ [1, ∞). The distribution of this product geometrically corresponds to the projection of the p-radial distribution on Bnp onto a random k-dimensional subspace. We derive large deviation principles (LDPs) on the space of probability measures on Rk for sequences of such projections.

Moderate Deviation and Large Deviation for Wegman-Davies Recursive Density Estimators

Miao Yu, Gao Qinghui, Mu Jianyong, Deng Conghui

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 1

83--95

Let {Xk, k ≥ 1} be a sequence of independent identically distributed random variables with common probability density function f, and let fn denote a Wegman-Davies recursive density estimator [formula] where K is a kernel function and hn is a band sequence. In the prezent paper, the moderate deviation principle and the large deviation principle for the estimator fn are established.

Large deviations for longest runs in Markov chains

Liu Zhenxia, Zhu Yurong

Journal of Applied Analysis

2020

Vol. 26, nr 2

309--314

We continue our investigation on general large deviation principles (LDPs) for longest runs. Previously, a general LDP for the longest success run in a sequence of independent Bernoulli trails was derived in [Z. Liu and X. Yang, A general large deviation principle for longest runs, Statist. Probab. Lett. 110 (2016), 128-132]. In the present note, we establish a general LDP for the longest success run in a two-state (success or failure) Markov chain which recovers the previous result in the aforementioned paper. The main new ingredient is to implement suitable estimates of the distribution function of the longest success run recently established in [Z. Liu and X. Yang, On the longest runs in Markov chains, Probab. Math. Statist. 38 (2018), no. 2, 407-428].

On the longest runs in Markov chains

Liu Z., Yang X.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 2

407--428

In the first n steps of a two-state (success and failure) Markov chain, the longest success run L(n) has been attracting considerable attention due to its various applications. In this paper, we study L(n) in terms of its two closely connected properties: moment generating function and large deviations. This study generalizes several existing results in the literature, and also finds an application in statistical inference. Our metod on the moment generating function is based on a global estimate of the cumulative distribution function of L(n) proposed in this paper, and the proofs of the large deviations include the Gärtner-Ellis theorem and the moment generating function.

Large deviations for wishart processes

Donati-Martin C.

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 2

325--343

Let Xδ be a Wishart process of dimension δ, with values in the set of positive matrices of size m. We are interested in the large deviations for a family of matrix-valued processes {δ−1X(δ)t ; t ≤1} as δ tends to infinity. The process X(δ) is a solution of a stochastic differential equation with a degenerate diffusion coefficient. Our approach is based upon the introduction of exponential martingales. We give some applications to large deviations for functionals of the Wishart processes, for example the set of eigenvalues.