Witold Bednorz

A Few Remarks on “Fixed-Width Output Analysis for Markov Chain Monte Carlo” by Jones et al

Our aim is to relax assumptions and simplify proofs in results given by Jones, Haran, Caffo, and Neath in the recent paper “Fixed-Width Output Analysis for Markov Chain Monte Carlo.”

A theorem on majorizing measures

Let (T,d) be a metric space and φ:R + → 1R an increasing, convex function with φ(0) = 0. We prove that if m is a probability measure m on r which is majorizing with respect to d, φ, that is, &:= supper ∫ D(T) 0 φ -1 (1 m(B(x,e)))de < ∞ then E sup |X(s) - X(t)|≤32&/s,t∈T for each separable stochastic process X(t), t ∈ T, which satisfies Eφ(|X(s)-X(t)|/d(s,t)) < 1 for all s,t ∈ T, s ≠ t. This is a strengthening of one of the main results from Talagrand [Ann. Probab. 18 (1990) 1-491, and its proof is significantly simpler.

Dynamics of a stochastic spatially extended system predicted by comparing deterministic and stochastic attractors of the corresponding birth-death process

Living cells may be considered as biochemical reactors of multiple steady states. Transitions between these states are enabled by noise, or, in spatially extended systems, may occur due to the traveling wave propagation. We analyze a one-dimensional bistable stochastic birth-death process by means of potential and temperature fields. The potential is defined by the deterministic limit of the process, while the temperature field is governed by noise. The stable steady state in which the potential has its global minimum defines the global deterministic attractor. For the stochastic system, in the low noise limit, the stationary probability distribution becomes unimodal, concentrated in one of two stable steady states, defined in this study as the global stochastic attractor. Interestingly, these two attractors may be located in different steady states. This observation suggests that the asymptotic behavior of spatially extended stochastic systems depends on the substrate diffusivity and size of the reactor. We confirmed this hypothesis within kinetic Monte Carlo simulations of a bistable reaction- diffusion model on the hexagonal lattice. In particular, we found that although the kinase-phosphatase system remains inactive in a small domain, the activatory traveling wave may propagate when a larger domain is considered.

On the boundedness of Bernoulli processes

We present a positive solution to the so-called Bernoulli Conjecture concerning the characterization of sample boundedness of Bernoulli processes. We also discuss some applications and related open problems.

L^1-smoothing for the Ornstein-Uhlenbeck semigroup

Given a probability density, we estimate the rate of decay of the measure of the level sets of its evolutes by the Ornstein–Uhlenbeck semigroup. The rate is faster than what follows from the preservation of mass and Markov’s inequality.

Orlicz Integrability of Additive Functionals of Harris Ergodic Markov Chains

For a Harris ergodic Markov chain (X n )n ≥ 0, on a general state space, started from the small measure or from the stationary distribution, we provide optimal estimates for Orlicz norms of sums ∑i = 0 τ f(X i ), where τ is the first regeneration time of the chain. The estimates are expressed in terms of other Orlicz norms of the function f (with respect to the stationary distribution) and the regeneration time τ (with respect to the small measure). We provide applications to tail estimates for additive functionals of the chain (X n ) generated by unbounded functions as well as to classical limit theorems (CLT, LIL, Berry-Esseen).

The complete characterization of a.s. convergence of orthogonal series

In this paper we prove the complete characterization of a.s. convergence of orthogonal series in terms of existence of a majorizing measure. It means that for a given (an)∞n=1, an>0, series ∑∞n=1anφn is a.e. convergent for each orthonormal sequence (φn)∞n=1 if and only if there exists a measure m on T={0}∪{∑n=1ma2n,m≥1} such that supt∈T∫D(T)√0(m(B(t,r2)))−1/2dr<∞, where D(T)=sups,t∈T|s−t| and B(t,r)={s∈T : |s−t|≤r}. The presented approach is based on weakly majorizing measures and a certain partitioning scheme.

Greedy Type Bases in Banach Spaces

Let (X, · ) be a (real) Banach space. We refer to [38] or [28] as some introduction to the general theory of Banach spaces. Note that, as usual in the case, all the results we discuss here remain valid for complex scalars with possibly different constants. Let I be a countable set with possibly some ordering we refer to whenever considering convergence with respect to elements of I (wich will be denoted by limi→∞). Definition 1 We say that countable system of vectors is biorthogonal if for i, j ∈ I we have

Bounds for Stochastic Processes on Product Index Spaces

In this paper we discuss the question of how to bound the supremum of a stochastic process with an index set of a product type. It is tempting to approach the question by analyzing the process on each of the marginal index sets separately. However it turns out that it is necessary to also study suitable partitions of the entire index set. We show what can be done in this direction and how to use the method to reprove some known results. In particular we point out that all known applications of the Bernoulli Theorem can be obtained in this way. Moreover we use the shattering dimension to slightly extend the application to VC classes. We also show some application to the regularity of paths of processes which take values in vector spaces. Finally we give a short proof of the Mendelson–Paouris result on sums of squares for empirical processes.

Testing locality and noncontextuality with the lowest moments

The quest for fundamental tests of quantum mechanics is an ongoing effort. We here address the question of what are the lowest possible moments needed to prove quantum nonlocality and noncontextuality without any further assumptions—in particular, without the often assumed dichotomy. We first show that second-order correlations can always be explained by a classical noncontextual local-hidden-variable theory. Similar third-order correlations also cannot violate classical inequalities in general, except for a special state-dependent noncontextuality. However, we show that fourth-order correlations can violate locality and state-independent noncontextuality. Finally we obtain a fourth-order continuous-variable Bell inequality for position and momentum, which can be violated and might be useful in Bell tests, closing all loopholes simultaneously.