Evgeny Baklanov
The strong law of large numbers for linear combinations of functions of order statistics ($L$-statistics) based on weakly dependent random variables is proven. We also establish the Glivenko--Cantelli theorem for $ϕ$-mixing sequences of identically distributed random variables.
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We study the law of the iterated logarithm (LIL) for the maximum likelihood estimation of the parameters (as a convex optimization problem) in the generalized linear models with independent or weakly dependent ($ρ$-mixing, $m$-dependent) responses under mild conditions. The LIL is useful to derive the asymptotic bounds for the discrepancy between the empirical process of the log-likelihood function and the true log-likelihood. As the application of the LIL, the strong consistency of some penalized likelihood based model selection criteria can be shown. Under some regularity conditions, the mod…
Functional central limit theorem and Marcinkiewicz strong law of large numbers for Hilbert-valued U-statistics of absolutely regular data
本文针对绝对正则的希尔伯特值U统计量，采用耦合方法推导含混合系数与i.i.d. U统计量尾部的偏差不等式，改进了泛函中心极限定理与Marcinkiewicz强大数定律的相关结果，放宽了混合速率限制并拓展了取值空间范围。
The class of Generalized $L$-statistics ($GL$-statistics) unifies a broad class of different estimators, for example scale estimators based on multivariate kernels. $GL$-statistics are functionals of $U$-quantiles and therefore the dimension of the kernel of the $U$-quantiles determines the kernel dimension of the estimator. Up to now only few results for multivariate kernels are known. Additionally, most theory was established under independence or for short range dependent processes. In this paper we establish a central limit theorem for $GL$-statistics of functionals of short range dependen…
In this manuscript, we study stochastic comparisons of the second-order statistics from dependent or independent observations with modified proportional hazard rates models. First, we establish the usual stochastic order of the second-order statistics from dependent and heterogeneous observations. Second, sufficient conditions are provided in the hazard rate order of the second-order statistics from independent observations. Then, we investigate the hazard rate order of the second-order statistics arising from two sets of independent multiple-outlier modified proportional hazard rates observat…
We consider the problem of conducting an experiment to study the prevalence of racial bias against individuals seeking legal assistance, in particular whether lawyers use clues about a potential client's race in deciding whether to reply to e-mail requests for representations. The problem of discriminating between potential linear and non-linear effects of a racial signal is formulated as a statistical inference problem, whose objective is to infer a parameter determining the shape of a specific function. Various complexities associated with the design and analysis of this experiment are handl…
Tunable robustness in power-law inference
针对幂律推断中测量误差导致统计偏差的问题，提出使用对数分箱和离散参考分布的方法，有效减弱了参数估计偏差并重新校准了拟合优度，以适度统计功效代价使实证结论更符合预期。
On the Law of Large Numbers for non-equally distributed weakly dependent random variables
本文研究弱相依、非等分布随机变量在有限或无限期望情形下的大数定律，给出三个版本的弱大数定律。
The law of the iterated logarithm for partial sums of weakly dependent processes was intensively studied by Walter Philipp in the late 1960s and 1970s. In this paper, we aim to extend these results to nondegenerate U-statistics of data that are strongly mixing or functionals of an absolutely regular process.
Strong law of large numbers for random walks in weakly dependent random scenery
本文在随机场景非平稳且满足适当速率弱相依条件的假设下，建立了随机游走在随机场景中的强大数定律，推广了已有文献结果。
We consider a class of real random matrices with dependent entries and show that the limiting empirical spectral distribution is given by the Marchenko-Pastur law. Additionally, we establish a rate of convergence of the expected empirical spectral distribution.
Strong laws of large numbers for sequences of blockwise $m$-dependent and orthogonal random variables under sublinear expectations
本文在次线性期望框架下，针对块wise $m$-相依和正交这两类非独立随机变量，建立了强大数定律，推广了该空间中独立随机变量的强大数定律。
In this paper we study ensembles of random symmetric matrices $\X_n = {X_{ij}}_{i,j = 1}^n$ with dependent entries such that $\E X_{ij} = 0$, $\E X_{ij}^2 = σ_{ij}^2$, where $σ_{ij}$ may be different numbers. Assuming that the average of the normalized sums of variances in each row converges to one and Lindeberg condition holds we prove that the empirical spectral distribution of eigenvalues converges to Wigner's semicircle law.
Under some mild regularity on the normalizing sequence, we obtain necessary and sufficient conditions for the Strong Law of Large Numbers for (symmetrized) U-statistics. We also obtain nasc's for the a.s. convergence of series of an analogous form.
On the Law of the Iterated Logarithm for m-dependent stationary random variables under sub-linear expectations
该研究在次线性期望框架下，先将重对数律推广至独立非同分布随机变量序列，再以此为中间步骤建立m-相依平稳序列的重对数律，并给出相应必要条件，拓展了次线性期望下的极限理论。
Over the last two decades, it has been argued that the Lorentz transformation mechanism, which imposes the generalization of Newton's classical mechanics into Einstein's special relativity, implies a generalization, or deformation, of the ordinary statistical mechanics. The exponential function, which defines the Boltzmann's factor, emerges properly deformed within this formalism. Starting from this, so-called $κ$-deformed exponential function, we introduce new classes of statistical distributions emerging as the $κ$-deformed version of already known distribution as the Generalized Gamma, Weib…
The occupation time of an age-dependent branching particle system in $\Rd$ is considered, where the initial population is a Poisson random field and the particles are subject to symmetric $α$-stable migration, critical binary branching and random lifetimes. Two regimes of lifetime distributions are considered: lifetimes with finite mean and lifetimes belonging to the normal domain of attraction of a $γ$-stable law, $γ\in(0,1)$. It is shown that in dimensions $d>αγ$ for the heavy-tailed lifetimes case, and $d>α$ for finite mean lifetimes, the occupation time proccess satisfies a strong law of l…
We formulate conditions for convergence of Laws of Large Numbers and show its links with of the parts of mathematical analysis such as summation theory, convergence of orthogonal series. We present also applications of the Law of Large Numbers such as Stochastic Approximation, Density and Regression Estimation, Identification.
Level numbers preserving transformations on excursionsdefined by random walks with state dependent jump laws
本文研究了状态相关跳跃律随机游走游程定义的层号，计算了固定层号的游程数量并证明平移类可生成所有同层号游程，同时分析了层号在 Vervaat 和 Doob 变换下的行为。
We give a short, self-contained, and elementary proof of the strong law of large numbers under a power law decay hypothesis for joint second moments. The result is related to the classical one by Lyons. However, we also provide a rate of convergence. Our proof does not use maximal inequalities and is instead inspired by the method of multiscale large versus small field decompositions in constructive quantum field theory.
We consider random hermitian matrices in which distant above-diagonal entries are independent but nearby entries may be correlated. We find the limit of the empirical distribution of eigenvalues by combinatorial methods. We also prove that the limit has algebraic Stieltjes transform by an argument based on dimension theory of noetherian local rings.
The main purpose of this paper is to obtain strong laws of large numbers for arrays or weighted sums of random variables under a scenario of dependence. Namely, for triangular arrays $\{X_{n,k}, \, 1 \leqslant k \leqslant n, \, n \geqslant 1 \}$ of row-wise extended negatively dependent random variables weakly mean dominated by a random variable $X \in \mathscr{L}_{1}$ and sequences $\{b_{n} \}$ of positive constants, conditions are given to ensure $\sum_{k=1}^{n} \left(X_{n,k} - \mathbb{E} \, X_{n,k} \right)/b_{n} \overset{\textnormal{a.s.}}{\longrightarrow} 0$. Our statements also allow us t…
Strong law of large numbers for $m$-dependent and stationary random variables under sub-linear expectations
本文在次线性期望框架下，先建立独立非同分布随机变量的强大数定律，再将其拓展至满足特定条件的$m$-相依平稳随机变量序列，推广了次线性期望下的大数定律理论。
When estimating the risk of a financial position with empirical data or Monte Carlo simulations via a tail-dependent law invariant risk measure such as the Conditional Value-at-Risk (CVaR), it is important to ensure the robustness of the statistical estimator particularly when the data contain noise. Kratscher et al. [1] propose a new framework to examine the qualitative robustness of estimators for tail-dependent law invariant risk measures on Orlicz spaces, which is a step further from earlier work for studying the robustness of risk measurement procedures by Cont et al. [2]. In this paper,…
By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.
This paper is concerned with the limit laws of the extreme order statistics derived from a symmetric Laplace walk. We provide two different descriptions of the point process of the limiting extreme order statistics: a branching representation and a squared Bessel representation. These complementary descriptions expose various hidden symmetries in branching processes and Brownian motion which lie behind some striking formulas found by Schehr and Majumdar (Phys. Rev. Lett., 108:040601). In particular, the Bessel process of dimension $4 = 2+2$ appears in the descriptions as a path decomposition o…
We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n-dimentional Hermitian matrices as n tends to infinity. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al for orthogonal polynomials with varying weights, we show first that if the support of the Density of States of the model consists of two or more intervals, then in the global regime the variance of statistics is a quasiperiodic function of n generically in the potential, determining the mo…
Every sequence $f_1, f_2, \cdots \, $ of random variables with $ \, \lim_{M \to \infty} \big( M \sup_{k \in \mathbb{N}} \mathbb{P} ( |f_k| > M ) \big)=0\,$ contains a subsequence $ f_{k_1}, f_{k_2} , \cdots \,$ that satisfies, along with all its subsequences, the weak law of large numbers: $ \, \lim_{N \to \infty} \big( (1/N) \sum_{n=1}^N f_{k_n} - D_N \big) =0\,,$ in probability. Here $\, D_N\, $ is a "corrector" random variable with values in $[-N,N]$, for each $N \in \mathbb{N} $; these correctors are all equal to zero if, in addition, $\, \liminf_{k \to \infty} \mathbb{E} \big( f_k^2 \, \m…
Many systems exhibit a digit bias. For example, the first digit base 10 of the Fibonacci numbers, or of $2^n$, equals 1 not 10% or 11% of the time, as one would expect if all digits were equally likely, but about 30% of the time. This phenomenon, known as Benford's Law, has many applications, ranging from detecting tax fraud for the IRS to analyzing round-off errors in computer science. The central question is determining which data sets follow Benford's law. Inspired by natural processes such as particle decay, our work examines models for the decomposition of conserved quantities. We prove t…
This paper studies a limit order book (LOB) model, in which the order dynamics depend on both, the current best available prices and the current volume density functions. For the joint dynamics of the best bid price, the best ask price, and the standing volume densities on both sides of the LOB we derive a weak law of large numbers, which states that the LOB model converges to a continuous-time limit when the size of an individual order as well as the tick size tend to zero and the order arrival rate tends to infinity. In the scaling limit the two volume densities follow each a non-linear PDE …
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