We propose strongly consistent estimators of the $ell_1$ norm of the sequence of $alpha$-mixing (respectively $beta$-mixing) coefficients of a stationary ergodic process. We further provide strongly consistent estimators of individual $alpha$-mixing
(respectively $beta$-mixing) coefficients for a subclass of stationary $alpha$-mixing (respectively $beta$-mixing) processes with summable sequences of mixing coefficients. The estimators are in turn used to develop strongly consistent goodness-of-fit hypothesis tests. In particular, we develop hypothesis tests to determine whether, under the same summability assumption, the $alpha$-mixing (respectively $beta$-mixing) coefficients of a process are upper bounded by a given rate function. Moreover, given a sample generated by a (not necessarily mixing) stationary ergodic process, we provide a consistent test to discern the null hypothesis that the $ell_1$ norm of the sequence $boldsymbol{alpha}$ of $alpha$-mixing coefficients of the process is bounded by a given threshold $gamma in [0,infty)$ from the alternative hypothesis that $leftlVert boldsymbol{alpha} rightrVert> gamma$. An analogous goodness-of-fit test is proposed for the $ell_1$ norm of the sequence of $beta$-mixing coefficients of a stationary ergodic process. Moreover, the procedure gives rise to an asymptotically consistent test for independence.
We study the number of facets of the convex hull of n independent standard Gaussian points in d-dimensional Euclidean space. In particular, we are interested in the expected number of facets when the dimension is allowed to grow with the sample size.
We establish an explicit asymptotic formula that is valid whenever d/n tends to zero. We also obtain the asymptotic value when d is close to n.
We study the problem of estimating the mean of a multivariatedistribution based on independent samples. The main result is the proof of existence of an estimator with a non-asymptotic sub-Gaussian performance for all distributions satisfying some mild moment assumptions.
We discuss the possibilities and limitations of estimating the mean of a real-valued random variable from independent and identically distributed observations from a non-asymptotic point of view. In particular, we define estimators with a sub-Gaussia
n behavior even for certain heavy-tailed distributions. We also prove various impossibility results for mean estimators.
We investigate the effective resistance $R_n$ and conductance $C_n$ between the root and leaves of a binary tree of height $n$. In this electrical network, the resistance of each edge $e$ at distance $d$ from the root is defined by $r_e=2^dX_e$ where
the $X_e$ are i.i.d. positive random variables bounded away from zero and infinity. It is shown that $mathbf{E}R_n=nmathbf{E}X_e-(operatorname {mathbf{Var}}(X_e)/mathbf{E}X_e)ln n+O(1)$ and $operatorname {mathbf{Var}}(R_n)=O(1)$. Moreover, we establish sub-Gaussian tail bounds for $R_n$. We also discuss some possible extensions to supercritical Galton--Watson trees.
It is shown that functions defined on ${0,1,...,r-1}^n$ satisfying certain conditions of bounded differences that guarantee sub-Gaussian tail behavior also satisfy a much stronger ``local sub-Gaussian property. For self-bounding and configuration fun
ctions we derive analogous locally subexponential behavior. The key tool is Talagrands [Ann. Probab. 22 (1994) 1576--1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on ${0,1,...,r-1}^n$ for $rge2$.
Discussion of ``2004 IMS Medallion Lecture: Local Rademacher complexities and oracle inequalities in risk minimization by V. Koltchinskii [arXiv:0708.0083]
