arXiv subjects

Thomas Löbbe

Publications and source records attributed to Thomas Löbbe.

4 recordsLinked to original sources

arXiv2014

Limit Theorems for Multivariate Lacunary Systems

Lacunary function systems of type $(f(M_nx))_{n\geq 1}$ for periodic functions $f$ and sequences of fast-growing matrices $(M_n)_{n\geq 1}$ exhibit many properties of independent random variables like satisfying the Central Limit Theorem or the Law of the Iterated Logarithm. It is well-known that this behaviour depends on number theoretic properties of $(M_n)_{n\geq 1}$ as well as analytic properties of $f$. Classical techniques are essentially based on Fourier analysis making it almost impossible to use a similar approach in the multivariate setting. Recently Aistleitner and Berkes introduced a new method proving the Central Limit Theorem in the one-dimensional case by approximating $\sum_{n}f(M_nx)$ by a sum of piecewise constant periodic functions which form a martingale differences sequence and using a Berry-Esseen type inequality. Later this approach was used to show the Law of the Iterated Logarithm by a consequence of Strassen's almost sure invariance principle. In this paper we develop this method to prove the Central Limit Theorem and the Law of the Iterated Logarithm in the multidimensional case.

math.PR↗

arXiv2014

Star Discrepancy Bounds of Double Infinite Matrices induced by Lacunary Systems

In 2001 Heinrich, Novak, Wasilkowski and Woźniakowski proved that the inverse of the star discrepancy satisfies $n(d,\varepsilon)\leq c_{\abs}d \varepsilon^{-2}$ by showing that there exists a set of points in $[0,1)^d$ whose star-discrepancy is bounded by $c_{\abs}\sqrt{d/N}$. This result was generalized by Aistleitner who showed that there exists a double infinite random matrix with elements in $[0,1)$ which partly are coordinates of elements of a Halton sequence and partly independent uniformly distributed random variables such that any $N\times d$-dimensional projection defines a set $\{x_1,\ldots,x_N\}\subset [0,1)^d$ with \begin{equation*} D^*_N(x_1,\ldots,x_N)\leq c_{\abs}\sqrt{d/N}. \end{equation*} In this paper we consider a similar double infinite matrix where the elements instead of independent random variables are taken from a certain multivariate lacunary sequence and prove that with high probability each projection defines a set of points which has up to some constant the same upper bound on its star-discrepancy but only needs a significantly lower number of digits to simulate.

math.PR↗

arXiv2014

Spectral Distribution of Non-independent Random Matrix Ensembles induced by Lacunary Systems

For two lacunary sequences $(M_{n,1})_{n\geq 2},(M_{n,2})_{n\geq 0}$ and suitable functions $f$ we introduce random matrix ensembles with \begin{equation*} X_{n,n'}=f(M_{n+n',1}x_1,M_{|n-n'|,2}x_2). \end{equation*} We prove weak convergence of the mean empirical eigenvalue distribution towards the semicircle law under some further number theoretic properties of the sequence $(M_{n,1})_{n\geq 1}$. Furthermore we give examples to show that even in this particular class of random matrix ensembles the asymptotic behaviour of the spectrum becomes delicate. We prove that the empirical spectral distribution does not converge to the semicircle law in general even if the correlation of two entries decays exponentially in the distance. For $f(x_1,x_2)=1/\sqrt{2}\cdot(\cos(2π(x_1+x_2))+\cos(4π(x_1+x_2)))$ and $M_{n,1}=2^n$ we show that the mean empirical spectral distribution does not converge to semicircle law while for any sequence $(M_{n,1})_{n\geq 1}$ with $M_{n+1,1}/M_{n,1}\to\infty$ for $n\to\infty$ and any periodic function $f$ of finite total variation in the sense of Hardy and Krause with mean zero and unit variance the mean spectral distribution converges to the semicircle law.

math.PR↗

arXiv2014

Probabilistic Star Discrepancy Bounds for Lacunary Point Sets

By a result of Heinrich, Novak, Wasilkowski and Woźniakowski the inverse of the star discrepancy $n(d,\varepsilon)$ satisfies $n(d,\varepsilon)\leq c_{\abs}d\varepsilon^{-2}$. Equivalently for any $N$ and $d$ there exists a set of $N$ points in $[0,1)^d$ with star discrepacny bounded by $\sqrt{c_{\abs}\cdot d/N}$. They actually proved that a set of independent uniformly distributed random points satisfies this upper bound with positive probability. Although Aistleitner and Hofer later refined this result by proving a precise value of $c_{\abs}$ depending on the probability with which the inequality holds, so far there is no general construction for such a set of points known. In this paper we consider the sequence $(x_n)_{n\geq 1}=(\langle 2^{n-1}x_1\rangle)_{n\geq 1}$ for a uniformly distributed point $x_1\in [0,1)^d$ and prove that the star discrepancy is bounded by $C\sqrt{d\log_2d/N}$. The precise value of $C$ depends on the probability with which this upper bound holds.

math.PR↗