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Mariusz Mirek

Publications and source records attributed to Mariusz Mirek.

45 records · Page 3Linked to original sources
arXiv2014

Roth's Theorem in the Piatetski-Shapiro primes

Let $\mathbf{P}$ denote the set of prime numbers and, for an appropriate function $h$, define a set $\mathbf{P}_{h}=\{p\in\mathbf{P}: \exists_{n\in\mathbb{N}}\ p=\lfloor h(n)\rfloor\}$. The aim of this paper is to show that every subset of $\mathbf{P}_{h}$ having positive relative upper density contains a nontrivial three-term arithmetic progression. In particular the set of Piatetski--Shapiro primes of fixed type $71/72<γ<1$, i.e. $\{p\in\mathbf{P}: \exists_{n\in\mathbb{N}}\ p=\lfloor n^{1/γ}\rfloor\}$ has this feature. We show this by proving the counterpart of Bourgain--Green's restriction theorem for the set $\mathbf{P}_{h}$.

math.CA↗
arXiv2014

Weak type (1, 1) inequalities for discrete rough maximal functions

The aim of this paper is to show that the discrete maximal function $$\mathcal{M}_{h}f(x)=\sup_{N\in\mathbb{N}}\frac{1}{|\mathbf{N}_{h}\cap[1, N]|}\Big|\sum_{n\in \mathbf{N}_{h}\cap[1, N]}f(x-n)\Big|,\ \ \mbox{for $x\in\mathbb{Z}$},$$ is of weak type $(1, 1)$, where $\mathbf{N}_{h}=\{n\in\mathbb{N}: \exists_{m\in\mathbb{N}}\ n=\lfloor h(m)\rfloor\}$ for an appropriate function $h$. As a consequence we also obtain pointwise ergodic theorem along the set $\mathbf{N}_{h}$.

math.CA↗
arXiv2013

Heavy tailed solutions of multivariate smoothing transforms

Let $N > 1$ be a fixed integer and $(C_1,..., C_N,Q)$ a random element of $GL(d, \R)^N x \R^d$. We consider solutions of multivariate smoothing transforms, i.e. random variables $R$ satisfying $$R \eqdist \sum_{i=1}^N C_i R_i +Q $$ where $\eqdist$ denotes equality in distribution, and $R, R_1,..., R_N$ are independent identically distributed $\R^d$-valued random variables, and independent of $(C_1,..., C_N, Q)$. We briefly review conditions for the existence of solutions, and then study their asymptotic behaviour. We show that under natural conditions, these solutions exhibit heavy tails. Our results also cover the case of complex valued weights $(C_1,..., C_N)$.

math.PR↗
arXiv2012

Convergence to stable laws for multidimensional stochastic recursions: the case of regular matrices

Given a sequence $(M_{n},Q_{n})_{n\ge 1}$ of i.i.d.\ random variables with generic copy $(M,Q) \in GL(d, \R) \times \R^d$, we consider the random difference equation (RDE) $$ R_{n}=M_{n}R_{n-1}+Q_{n}, $$ $n\ge 1$, and assume the existence of $κ>0$ such that $$ \lim_{n \to \infty}(\E{\norm{M_1 ... M_n}^κ})^{\frac{1}{n}} = 1 .$$ We prove, under suitable assumptions, that the sequence $S_n = R_1 + ... + R_n$, appropriately normalized, converges in law to a multidimensional stable distribution with index $κ$. As a by-product, we show that the unique stationary solution $R$ of the RDE is regularly varying with index $κ$, and give a precise description of its tail measure. This extends the prior work http://arxiv.org/abs/1009.1728v3 .

math.PR↗
arXiv2011

On fixed points of a generalized multidimensional affine recursion

Let $G$ be a multiplicative subsemigroup of the general linear group $\Gl(\mathbb{R}^d)$ which consists of matrices with positive entries such that every column and every row contains a strictly positive element. Given a $G$--valued random matrix $A$, we consider the following generalized multidimensional affine equation R\stackrel{\mathcal{D}}{=}\sum_{i=1}^N A_iR_i+B, where $N\ge2$ is a fixed natural number, $A_1,...,A_N$ are independent copies of $A$, $B\in\mathbb{R}^d$ is a random vector with positive entries, and $R_1,...,R_N$ are independent copies of $R\in\mathbb{R}^d$, which have also positive entries. Moreover, all of them are mutually independent and $\stackrel{\mathcal{D}}{=}$ stands for the equality in distribution. We will show with the aid of spectral theory developed by Guivarc'h and Le Page and Kesten's renewal theorem, that under appropriate conditions, there exists $χ>0$ such that $¶(\{ >t\})\asymp t^{-χ},$ as $t\to\8$, for every unit vector $u\in\mathbb{S}^{d-1}$ with positive entries.

math.PR↗
arXiv2011

Asymptotics of stationary solutions of multivariate stochastic recursions with heavy tailed inputs and related limit theorems

Let $Φ_n$ be an i.i.d. sequence of Lipschitz mappings of $\R^d$. We study the Markov chain $\{X_n^x\}_{n=0}^\infty$ on $\R^d$ defined by the recursion $X_n^x = Φ_n(X^x_{n-1})$, $n\in\N$, $X_0^x=x\in\R^d$. We assume that $Φ_n(x)=Φ(A_n x,B_n(x))$ for a fixed continuous function $Φ:\R^d\times \R^d\to\R^d$, commuting with dilations and i.i.d random pairs $(A_n,B_n)$, where $A_n\in {End}(\R^d)$ and $B_n$ is a continuous mapping of $\R^d$. Moreover, $B_n$ is $α$-regularly varying and $A_n$ has a faster decay at infinity than $B_n$. We prove that the stationary measure $ν$ of the Markov chain $\{X_n^x\}$ is $α$-regularly varying. Using this result we show that, if $α<2$, the partial sums $S_n^x=\sum_{k=1}^n X_k^x$, appropriately normalized, converge to an $α$-stable random variable. In particular, we obtain new results concerning the random coefficient autoregressive process $X_n = A_n X_{n-1}+B_n$.

math.PR↗
arXiv2010

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain $\{X_n^x\}_{n=0}^\infty$ on $\R^d$ defined by the stochastic recursion $X_{n}^{x}=\p_{θ_{n}}(X_{n-1}^{x})$, starting at $x\in\R^d$, where $θ_{1}, θ_{2},...$ are i.i.d. random variables taking their values in a metric space $(Θ, \mathfrak{r}),$ and $\p_{θ_{n}}:\R^d\mapsto\R^d$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure $ν$. Under appropriate assumptions on $\p_{θ_n}$, we will show that the measure $ν$ has a heavy tail with the exponent $α>0$ i.e. $ν(\{x\in\R^d: |x|>t\})\asymp t^{-α}$. Using this result we show that properly normalized Birkhoff sums $S_n^x=\sum_{k=1}^n X_k^x$, converge in law to an $α$--stable law for $α\in(0, 2]$.

math.PR↗