Mariusz Mirek

Discrete maximal functions in higher dimensions and applications to ergodic theory

We establish a higher dimensional counterpart of Bourgains pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates

HEAVY TAILED SOLUTIONS OF MULTIVARIATE SMOOTHING TRANSFORMS

V_r

Cotlar’s Ergodic Theorem Along the Prime Numbers

on

VARIATIONAL ESTIMATES FOR AVERAGES AND TRUNCATED SINGULAR INTEGRALS ALONG THE PRIME NUMBERS

L^p

Roth's theorem in the Piatetski-Shapiro primes

spaces for all

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

1<p<\infty

Square function estimates for discrete Radon transforms

and

Convergence to Stable Laws for Multidimensional Stochastic Recursions: The Case of Regular Matrices

r>\max\{p, p/(p-1)\}

On Dimension-Free Variational Inequalities for Averaging Operators in \({\mathbb{R}^d}\)

. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.

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

Let N>1 be a fixed integer and (C1,…,CN,Q) a random element of M(d×d,R)N×Rd. We consider solutions of multivariate smoothing transforms, i.e. random variables R satisfying R=d∑i=1NCiRi+Q where =d denotes equality in distribution, and R,R1,…,RN are independent identically distributed Rd-valued random variables, and independent of (C1,…,CN,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 (C1,…,CN).