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F. Götze

Publications and source records attributed to F. Götze.

At least 37 records · Page 2Linked to original sources
arXiv2015

Rate of Convergence of the Empirical Spectral Distribution Function to the Semi-Circular Law

Let $\mathbf X=(X_{jk})_{j,k=1}^n$ denote a Hermitian random matrix with entries $X_{jk}$, which are independent for $1\le j\le k\le n$. We consider the rate of convergence of the empirical spectral distribution function of the matrix $\mathbf W=\frac1{\sqrt n}\mathbf X$ to the semi-circular law assuming that $\mathbf E X_{jk}=0$, $\mathbf E X_{jk}^2=1$ and uniformly bounded eight moments. By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix $\mathbf W$ and the semi--circular law is of order $O(n^{-1}\log^{5}n)$ with high probability.

math.PR↗
arXiv2015

Asymptotic Expansions in Free Limit Theorems

We study asymptotic expansions in free probability. In a class of classical limit theorems Edgeworth expansion can be obtained via a general approach using sequences of "influence" functions of individual random elements described by vectors of real parameters $(\varepsilon_1,..., \varepsilon_n)$, that is by a sequence of functions $h_n(\varepsilon_1,..., \varepsilon_n;t)$, $|\varepsilon_j| \leq \frac 1 {\sqrt n}$, $j=1,...,n$, $t\in {\mathbb R}$ (or ${\mathbb C}$) which are smooth, symmetric, compatible and have vanishing first derivatives at zero. In this work we expand this approach to free probability. As a sequence of functions $h_n(\varepsilon_1,..., \varepsilon_n;t)$ we consider a sequence of the Cauchy transforms of the sum $\sum_{j=1}^n \varepsilon_j X_j $, where $(X_j)_{j=1}^n$ are free identically distributed random variables with nine moments. We derive Edgeworth type expansions for distributions and densities (under the additional assumption that ${supp} (X_1) \subset [-\sqrt [3]n, \sqrt [3]n]$) of the sum $\frac 1 {\sqrt n} \sum_{j=1}^n X_j$ within the interval $(-2,2)$.

math.PR↗
arXiv2014

On the Rate of Convergence to the Marchenko--Pastur Distribution

Let $\mathbf X=(X_{jk})$ denote $n\times p$ random matrix with entries $X_{jk}$, which are independent for $1\le j\le n,1\le k\le p$. We consider the rate of convergence of empirical spectral distribution function of the matrix $\mathbf W=\frac1p\mathbf X\mathbf X^*$ to the Marchenko--Pastur law. We assume that $\mathbf E X_{jk}=0$, $\mathbf E X_{jk}^2=1$ and that the distributions of the matrix elements $X_{jk}$ have a uniformly sub exponential decay in the sense that there exists a constant $\varkappa>0$ such that for any $1\le j \le n,\,1\le k\le p $ and any $t\ge 1$ we have $$ \mathbf{ Pr}\{|X_{jk}|>t\}\le \varkappa^{-1}\exp\{-t^{\varkappa}\}. $$ By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the sample covariance matrix $\mathbf W$ and the Marchenko--Pastur distribution is of order $O(n^{-1}\log^{4+\frac4{\varkappa}} n)$ with high probability.

math.PR↗
arXiv2014

Rate of Convergence of the Expected Spectral Distribution Function to the Marchenko -- Pastur Law

Let $\mathbf X=(X_{jk})$ denote a $n\times p$ random matrix with entries $X_{jk}$, which are independent for $1\le j\le n, 1\le k\le p$. Let $n,p$ tend to infinity such that $\frac np=y+O(n^{-1})\in(0,1]$. For those values of $n,p$ we investigate the rate of convergence of the expected spectral distribution function of the matrix $\mathbf W=\frac1{ p}\mathbf X\mathbf X^*$ to the Marchenko-Pastur law with parameter $y$. Assuming the conditions $\mathbf E X_{jk}=0$, $\mathbf E X_{jk}^2=1$ and $ \quad \quad \quad \quad \quad \quad \quad \sup_{n,p\ge1}\sup_{1\le j\le n,1\le k\le p}\mathbf E |X_{jk}|^4=: μ_4<\infty,\quad \sup_{n,p\ge1} \sup_{1\le j\le n,1\le k\le p}|X_{jk}|\le D n^{\frac14},$ we show that the Kolmogorov distance between the expected spectral distribution of the sample covariance matrix $\mathbf W$ and the Marchenko -- Pastur law is of order $O(n^{-1})$.

math.PR↗
arXiv2014

Asymptotic Spectra of Matrix-Valued Functions of Independent Random Matrices and Free Probability

We investigate the universality of singular value and eigenvalue distributions of matrix valued functions of independent random matrices and apply these general results in several examples. In particular we determine the limit distribution and prove universality under general conditions for singular value and eigenvalue distributions of products of independent matrices from spherical ensembles.

math.PR↗
arXiv2014

A multiplicative inequality for concentration functions of $n$-fold convolutions

We estimate the concentration functions of $n$-fold convolutions of one-dimensional probability measures. The main result is a supplement to the results of Götze and Zaitsev (1998). We show that the estimation of concentration functions at arguments of bounded size can be reduced to the estimation of these functions at arguments of size $O(\sqrt n)$ which is easier.

math.PR↗
arXiv2014

Multidimensional Hungarian construction for vectors with almost Gaussian smooth distributions

A multidimensional version of the results of Komlós, Major and Tusnády for sums of independent random vectors with finite exponential moments is obtained in the particular case where the summands have smooth distributions which are close to Gaussian ones. The bounds obtained reflect this closeness. Furthermore, the results provide sufficient conditions for the existence of i.i.d. vectors $X_1, X_2,\dots$ with given distributions and corresponding i.i.d. Gaussian vectors $Y_1, Y_2,\dots$ such that, for given small $\varepsilon$, $$ {\mathbf P}\Big\{{\limsup\limits_{n\to\infty} \frac1{\log n}\Bigl|\,\sum\limits_{j=1}^n X_j- \sum\limits_{j=1}^n Y_j\,\Bigr|}\le \varepsilon\Big\}=1. $$

math.PR↗
arXiv2013

The asymptotic number of integral cubic polynomials with bounded heights and discriminants

In the paper we partially solved the problem of the distribution of the discriminants of integral polynomials in the cubic case. We proved the asymptotic formula for the number of integral cubic polynomials having bounded height and bounded discriminant. The main term of the obtained formula is the best possible for the considered range of polynomial discriminant values.

math.NT↗
arXiv2013

Semicircle Law for a Class of Random Matrices with Dependent Entries

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.

math.PR↗
arXiv2012

Fisher information and the central limit theorem

An Edgeworth-type expansion is established for the relative Fisher information distance to the class of normal distributions of sums of i.i.d. random variables, satisfying moment conditions. The validity of the central limit theorem is studied via properties of the Fisher information along convolutions.

math.PR↗
arXiv2011

Asymptotic Expansions in the CLT in Free Probability

We prove Edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions. The proofs are based on the analytic definition of free convolution. We apply these results to the expansion of densities to derive expansions for the free entropic distance of sums to the Wigner law.

math.PR↗
arXiv2011

Rate of Convergence in the entropic free CLT

We prove an expansion for densities in the free CLT and apply this result to an expansion in the entropic free central limit theorem assuming a moment condition of order 8 for the free summands.

math.PR↗
arXiv2011

Characterization problems for linear forms with free summands

Let $T_1,...,T_n$ denote free random variables. For two linear forms $L_1=\sum_{j=1}^n a_jT_j$ and $L_2=\sum_{j=1}^n b_jT_j$ with real coefficients $a_j$ and $b_j$ we shall describe all distributions of $T_1,...,T_n$ such that $L_1$ and $L_2$ are free. For identically distributed free random variables $T_1,...,T_n$ with distribution $μ$ we establish necessary and sufficient conditions on the coefficients $a_j,b_j,\,j=1,...,n,$ such that the statements:\quad $(i)$ $μ$ is a centered semicircular distribution; and $(ii)$ \, $L_1$ and $L_2$ are identically distributed ($L_1\stackrel{D}{=}L_2$); are equivalent. In the proof we give a complete characterization of all sequences of free cumulants of measures with compact support and with a finite number of non zero entries. The characterization is based on topological properties of regions defined by means of the Voiculescu transform $ϕ$ of such sequences.

math.PR↗