arXiv subjects

Krzysztof Zajkowski

Publications and source records attributed to Krzysztof Zajkowski.

15 recordsLinked to original sources

arXiv2025

Concentration of norms of random vectors with independent $p$-sub-exponential coordinates

We present examples of $p$-sub-exponential random variables for any positive $p$. We prove two types of concentration of standard $p$-norms ($2$-norm is the Euclidean norm) of random vectors with independent $p$-sub-exponential coordinates around the Lebesgue $L^p$-norms of these $p$-norms of random vectors. In the first case $p\ge 1$, our estimates depend on the dimension $n$ of random vectors. But in the second one for $p\ge 2$, with an additional assumption, we get an estimate that does not depend on $n$. In other words, we generalize some know concentration results in the Euclidean case to cases of the $p$-norms of random vectors with independent $p$-sub-exponential coordinates.

math.PR↗

arXiv2024

Multivariate $α$-normal distributions

The Weibull distribution can be obtained using a power transformation from the standard exponential distribution. In this article, we will consider a symmetrized power transformation of a random variable with the standard normal distribution. We will call its distribution the $α$-{\it normal (Gaussian) distribution}. We examine properties of this distribution in detail. We calculate moments and consider the moment problem of $α$-normal distribution. We derive the formula of its differential entropy and (exponential) Orlicz norm. % of $α$-normal random variables. Moreover, we define the joint distribution function of the multivariate $α$-normal distribution as a meta-Gaussian distribution with $α$-normal marginals. We consider also the limiting distribution as $α$ tends to infinity.

math.PR↗

arXiv2021

On the strong law of large numbers for $φ$-subgaussian random variables

For $p\ge 1$ let $φ_p(x)=x^2/2$ if $|x|\le 1$ and $φ_p(x)=1/p|x|^p-1/p+1/2$ if $|x|>1$. For a random variable $ξ$ let $τ_{φ_p}(ξ)$ denote $\inf\{a\ge 0:\;\forall_{λ\in\mathbb{R}}\; \ln\mathbb{E}\exp(λξ)\leφ_p(aλ)\}$; $τ_{φ_p}$ is a norm in a space $Sub_{φ_p}=\{ξ:\;τ_{φ_p}(ξ)<\infty\}$ of $φ_p$-subgaussian random variables. We prove that if for a sequence $(ξ_n)\subset Sub_{φ_p}$ ($p>1$) there exist positive constants $c$ and $α$ such that for every natural number $n$ the following inequality $τ_{φ_p}(\sum_{i=1}^nξ_i)\le cn^{1-α}$ holds then $n^{-1}\sum_{i=1}^nξ_i$ converges almost surely to zero as $n\to\infty$. This result is a generalization of the SLLN for independent subgaussian random variables (Taylor and Hu \cite{TayHu}) to the case of dependent $φ_p$-subgaussian random variables.

math.PR↗

arXiv2020

Bounds on tail probabilities for quadratic forms in dependent sub-gaussian random variables

We show bounds on tail probabilities for quadratic forms in sub-gaussian non-necessarily independent random variables. Our main tool will be estimates of the Luxemburg norms of such forms. This will allow us to formulate the above-mentioned bounds. As an example we give estimates of the excess loss in fixed design linear regression in dependent observations.

math.PR↗

arXiv2019

On norms in some class of exponential type Orlicz spaces of random variables

A new characterization of the exponential type Orlicz spaces generated by the functions $\exp(|x|^p)-1$ ($p\ge 1$) is given. We define norms for centered random variables belonging to these spaces. We show equivalence of these norms with the Luxemburg norms. On the example of the Hoeffding inequality we present some application of these norms in a probabilistic context.

math.PR↗

arXiv2019

Norms of sub-exponential random vectors

We discuss various forms of the Luxemburg norm in spaces of random vectors with coordinates belonging to the classical Orlicz spaces of exponential type. We prove equivalent relations between some kinds of these forms. We also show when the so-called uniform norm is majorized by norms of coordinates up to some constants. We give an application of other norm to study of chaos in random vectors with sub-exponential coordinates.

math.PR↗

arXiv2017

A variational formula on the Cramér function of series of independent random variables

In [11] it has been proved some variational formula on the Legendre-Fenchel transform of the cumulant generating function (the Cramér function) of Rademacher series with coefficients in the space $\ell^1$. In this paper we show a generalization of this formula to series of a larger class of any independent random variables with coefficients that belong to the space $\ell^2$.

math.PR↗

arXiv2017

On the Azuma inequality in spaces of subgaussian of rank $p$ random variables

For $p > 1$ let a function $φ_p(x) = x^2/2$ if $|x|\le 1$ and $φ_p(x) = 1/p|x|^p -1/p + 1/2$ if $|x| > 1$. For a random variable $ξ$ let $τ_{φ_p}(ξ)$ denote $\inf\{c\ge 0 :\; \forall_{λ\in\mathbb{R}}\; \ln\mathbb{E}\exp(λξ)\leφ_p(cλ)\}$; $τ_{φ_p}$ is a norm in a space $Sub_{φ_p}(Ω) =\{ξ: \; τ_{φ_p}(ξ) <\infty\}$ of $φ_p$-subgaussian random variables which we call {\it subgaussian of rank $p$ random variables}. For $p = 2$ we have the classic subgaussian random variables. The Azuma inequality gives an estimate on the probability of the deviations of a zero-mean martingale $(ξ_n)_{n\ge 0}$ with bounded increments from zero. In its classic form is assumed that $ξ_0 = 0$. In this paper it is shown a version of the Azuma inequality under assumption that $ξ_0$ is any subgaussian of rank $p$ random variable.

math.PR↗

arXiv2014

Cramér transform of Rademacher series

A variational formula for the Cramér transform of series of weighted, independent symmetric Bernoulli random variables (Rademacher series) is given.

math.PR↗

arXiv2014

Penney's game between many players

We recall a combinatorial derivation of the functions generating probability of winnings for each of many participants of the Penney's game and show a generalization of the Conway's formula to this case.

math.PR↗

arXiv2014

Fejer's approximation of continuous functions of unitary operators

This paper is concerned with a certain aspect of the spectral theory of unitary operators in a Hilbert space and its aim is to give an explicit construction of continuous functions of unitary operators. Starting from a given unitary operator we give a family of sequences of trigonometric polynomials converging weakly to the complex measures which allow us to define functions of the operator.

math.FA↗

arXiv2013

A note on the gambling team method

Gerber and Li in \cite{GeLi} formulated, using a Markov chain embedding, a system of equations that describes relations between generating functions of waiting time distributions for occurrences of patterns in a sequence of independent repeated experiments when initial outcomes of the process are known. We show how this system of equations can be obtained by using the classical gambling team technique . We also present a form of solution of the system and give an example showing how first results of trials influence the probabilities that a chosen pattern precedes remaining ones in a realization of the process.

math.PR↗

arXiv2013

Cramér transform and t-entropy

t-entropy is the convex conjugate of the logarithm of the spectral radius of a weighted composition operator (WCO). Let $X$ be a nonnegative random variable. We show how the Cramér transform with respect to the spectral radius of WCO is expressed by the t-entropy and the Cramér transform of the given random variable X.

math.PR↗

arXiv2013

On the waiting time till some patterns occur in i.i.d. sequences

In this paper we present some general solution of the system of linear equations formed by Guibas and Odlyzko in Th.3.3 \cite{Gui}. We derive probabilities for given patterns to be first to appear in random text and the expected waiting time till one of them is observed and also till one of them occurs given it is known which pattern appears first.

math.PR↗

arXiv2011

Convex conjugates of analytic functions of logarithmically convex functionals

Let $f_{\bf c}(r)=\sum_{n=0}^\infty e^{c_n}r^n$ be an analytic function; ${\bf c}=(c_n)\in l_\infty$. We assume that $r$ is some logarithmically convex and lower semicontinuous functional on a locally convex topological space $L$. In this paper we derive a formula on the Legendre-Fenchel transform of a functional $\hatλ({\bf c},ϕ)=\ln f_{\bf c}(e^{λ(ϕ)})$, where $λ(ϕ)=\ln r(ϕ)$ ($ϕ\in L$). In this manner we generalize to the infinite case Theorem 3.1 from \cite{OZ1}.

math.FA↗