arXiv subjects

Joscha Prochno

Publications and source records attributed to Joscha Prochno.

81 records · Page 5Linked to original sources

arXiv2013

On the Distribution of Random variables corresponding to Musielak-Orlicz norms

Given a normalized Orlicz function $M$ we provide an easy formula for a distribution such that, if $X$ is a random variable distributed accordingly and $X_1,...,X_n$ are independent copies of $X$, then the expected value of the p-norm of the vector $(x_iX_i)_{i=1}^n$ is of the order $\| x \|_M$ (up to constants dependent on p only). In case $p=2$ we need the function $t\mapsto tM'(t) - M(t)$ to be 2-concave and as an application immediately obtain an embedding of the corresponding Orlicz spaces into $L_1[0,1]$. We also provide a general result replacing the $\ell_p$-norm by an arbitrary $N$-norm. This complements some deep results obtained by Gordon, Litvak, Schütt, and Werner. We also prove a result in the spirit of their work which is of a simpler form and easier to apply. All results are true in the more general setting of Musielak-Orlicz spaces.

math.FA↗

arXiv2013

Mean Width of Random Perturbations of Random Polytopes

We prove some "high probability" results on the expected value of the mean width for random perturbations of random polytopes. The random perturbations are considered for Gaussian and $p$-stable random vectors, as well as uniform distributions on $\ell_p^N$-balls and the unit sphere.

math.FA↗

arXiv2012

On mean outer radii of random polytopes

In this paper we introduce a new sequence of quantities for random polytopes. Let $K_N=\conv\{X_1,...,X_N\}$ be a random polytope generated by independent random vectors uniformly distributed in an isotropic convex body $K$ of $\R^n$. We prove that the so-called $k$-th mean outer radius $\widetilde R_k(K_N)$ has order $\max\{\sqrt{k},\sqrt{\log N}\}L_K$ with high probability if $n^2\leq N\leq e^{\sqrt{n}}$. We also show that this is also the right order of the expected value of $\widetilde R_k(K_N)$ in the full range $n\leq N\leq e^{\sqrt{n}}$.

math.FA↗

arXiv2012

A Combinatorial Approach to Musielak-Orlicz Spaces

In this paper we show that, using combinatorial inequalities and Matrix-Averages, we can generate Musielak-Orlicz spaces, i.e., we prove that $1/π\sum_π \max\limits_{1 \leq i \leq n} \abs{x_i y_{iπ(i)}} \sim \norm{x}_{ΣM_i}$, where the Orlicz functions $M_1,...,M_n$ depend on the matrix $(y_{ij})_{i,j=1}^n$. We also provide an approximation result for Musielak-Orlicz norms which already in the case of Orlicz spaces turned out to be very useful.

math.FA↗

arXiv2012

On the Gaussian behavior of marginals and the mean width of random polytopes

We show that the expected value of the mean width of a random polytope generated by $N$ random vectors ($n\leq N\leq e^{\sqrt n}$) uniformly distributed in an isotropic convex body in $\R^n$ is of the order $\sqrt{\log N} L_K$. This completes a result of Dafnis, Giannopoulos and Tsolomitis. We also prove some results in connection with the 1-dimensional marginals of the uniform probability measure on an isotropic convex body, extending the interval in which the average of the distribution functions of those marginals behaves in a sub- or supergaussian way.

math.FA↗

arXiv2012

Estimating Support Functions of Random Polytopes via Orlicz Norms

We study the expected value of support functions of random polytopes in a certain direction, where the random polytope is given by independent random vectors uniformly distributed in an isotropic convex body. All results are obtained by an utterly novel approach, using probabilistic estimates in connection with Orlicz norms that were not used in this connection before.

math.FA↗

arXiv2012

Combinatorial Inequalities and Subspaces of L1

Let M and N be Orlicz functions. We establish some combinatorial inequalities and show that the product spaces l^n_M(l^n_N) are uniformly isomorphic to subspaces of L_1 if M and N are "separated" by a function t^r, 1<r<2.

math.FA↗

arXiv2012

An Inversion Formula for Orlicz Norms and Sequences of Random Variables

Given an Orlicz function $M$, we show which random variables $ξ_i$, $i=1,...,n$ generate the associated Orlicz norm, i.e., which random variables yield $\mathbb{E} \max\limits_{1\leq i \leq n}|x_iξ_i| \sim \norm{(x_i)_{i=1}^n}_M$. As a corollary we obtain a representation for the distribution function in terms of $M$ and $M'$ which can be easily applied to many examples of interest.

math.FA↗

arXiv2012

On the Maximum of Random Variables on Product Spaces

Let $ξ_i$, $i=1,...,n$, and $η_j$, $j=1,...,m$ be iid p-stable respectively q-stable random variables, $1<p<q<2$. We prove estimates for $\Ex_{Ω_1} \Ex_{Ω_2}\max_{i,j}\abs{a_{ij}ξ_i(ω_1)η_j(ω_2)}$ in terms of the $\ell_p^m(\ell_q^n)$-norm of $(a_{ij})_{i,j}$. Additionally, for p-stable and standard gaussian random variables we prove estimates in terms of the $\ell_p^m(\ell_{M_ξ}^n)$-norm, $M_ξ$ depending on the Gaussians. Furthermore, we show that a sequence $ξ_i$, $i=1,...,n$ of iid $\log-γ(1,p)$ distributed random variables ($p\geq 2$) generates a truncated $\ell_p$-norm, especially $\Ex \max_{i}\abs{a_iξ_i}\sim \norm{(a_i)_i}_2$ for $p=2$. As far as we know, the generating distribution for $\ell_p$-norms with $p\geq 2$ has not been known up to now.

math.FA↗