Pawel Hitczenko

Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities

The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the p moment of a sum of independent symmetric random variables to that of the p and 2 moments of the individual variables, are computed in the range 2 < p ≤ 4. This complements the work of Utev who has done the same for p > 4. The qualitative nature of the extreme cases turns out to be different for p < 4 than for p > 4. The method developed yields results in some more general and other related moment inequalities.

Bursting Oscillations Induced by Small Noise

We consider a model of a square-wave bursting neuron residing in the regime of tonic spiking. Upon introduction of small stochastic forcing, the model generates irregular bursting. The statistical properties of the emergent bursting patterns are studied in the present work. In particular, we identify two principal statistical regimes associated with the noise-induced bursting. In the first case, type I, bursting oscillations are created mainly due to the fluctuations in the fast subsystem. In the alternative scenario, type II bursting, the random perturbations in the slow dynamics play a dominant role. We propose two classes of randomly perturbed slow-fast systems that realize type I and type II scenarios. For these models, we derive the Poincare maps. The analysis of the linearized Poincare maps of the randomly perturbed systems explains the distributions of the number of spikes within one burst and reveals their dependence on the small and control parameters present in the models. The mathematical analy...

Domination inequality for martingale transforms of a Rademacher sequence

AbstractLetfn = Σk=1nvkrk,n=1,…, be a martingale transform of a Rademacher sequence (rn)and let (rn′) be an independent copy of (rn).The main result of this paper states that there exists an absolute constantK such that for allp, 1≤p<∞, the following inequality is true:

Gap-free compositions and gap-free samples of geometric random variables

\left\| {\sum {v_k r_k } } \right\|_p \leqslant K\left\| {\sum {v_k r_k^\prime } } \right\|_p

Characterizations of orthonormal scale functions: A probabilistic approach

In order to prove this result, we obtain some inequalities which may be of independent interest. In particular, we show that for every sequence of scalars (an)one has

Note: Gaps in samples of geometric random variables

\left\| {\sum {v_k r_k } } \right\|_p \approx K_{1,2} ((a_n )),\sqrt p