Hanna Wojewódka-Ściążko

We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity λ. The model of this type, although in a more general version, was examined in our previous papers…

We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity λ. The model of this type, although in a more general version, was examined in our previous papers, where we have shown, among others, that the Markov process under consideration possesses a unique invariant probability measure, say ν(λ). The aim of this paper is to prove that the map λ↦ν(λ) is continuous (in the topology of weak convergence of probability measures). The studied dynamical system is inspired by certain stochastic models for cell division and gene expression. Show less

In the paper we propose conditions ensuring the central limit theorem (CLT) and the law of the iterated logarithm (LIL) for a certain class of Markov chains. We further use this general criteria to verify the aforementioned limit theorems for a particular disrete-time Markov system. The piecewise-deterministic Markov process defined via interpolation of the explored Markov chain can be used e.g. to describe a model for gene expression. The aim for the future work is to establish the CLT and the…

In the paper we propose conditions ensuring the central limit theorem (CLT) and the law of the iterated logarithm (LIL) for a certain class of Markov chains. We further use this general criteria to verify the aforementioned limit theorems for a particular disrete-time Markov system. The piecewise-deterministic Markov process defined via interpolation of the explored Markov chain can be used e.g. to describe a model for gene expression. The aim for the future work is to establish the CLT and the LIL for the continuous-time process too. Show less

The main goal of this paper is to establish the strong law of large numbers (SLLN) for a subclass of piecewise-deterministic Markov processes (PDMPs). On the way to this result, we provide sufficient conditions for the existence of an exponentially attracting invariant distribution for the Markov chain given by the post-jump locations of a PDMP. Furthermore, we obtain a one-to-one correspondence between invariant measures of such a chain and invariant measures of the PDMP. Finally, we…

The main goal of this paper is to establish the strong law of large numbers (SLLN) for a subclass of piecewise-deterministic Markov processes (PDMPs). On the way to this result, we provide sufficient conditions for the existence of an exponentially attracting invariant distribution for the Markov chain given by the post-jump locations of a PDMP. Furthermore, we obtain a one-to-one correspondence between invariant measures of such a chain and invariant measures of the PDMP. Finally, we illustrate the applicability of our results for a model of prokaryotic gene expression. Show less

The problem of device-independent randomness amplification against no-signaling adversaries has so far been studied under the assumption that the weak source of randomness is uncorrelated with the (quantum) devices used in the amplification procedure. In this paper, we relax this assumption, and reconsider the original protocol of Colbeck and Renner using a Santha-Vazirani (SV) source. To do so, we introduce an SV-like condition for devices, namely that any string of SV source bits remains…

The problem of device-independent randomness amplification against no-signaling adversaries has so far been studied under the assumption that the weak source of randomness is uncorrelated with the (quantum) devices used in the amplification procedure. In this paper, we relax this assumption, and reconsider the original protocol of Colbeck and Renner using a Santha-Vazirani (SV) source. To do so, we introduce an SV-like condition for devices, namely that any string of SV source bits remains weakly random conditioned upon any other bit string from the same SV source and the outputs obtained when this further string is input into the devices. Assuming this condition, we show that a quantum device using a singlet state to violate the chained Bell inequalities leads to full randomness in the asymptotic scenario of a large number of settings, for a restricted set of SV sources (with 0 ≤ ε <; (2 (1/12) - 1)/(2(2 (1/12) + 1)) ≈ 0.0144). We also study a device-independent protocol that allows for correlations between the sequence of boxes used in the protocol and the SV source bits used to choose the particular box from whose output the randomness is obtained. Assuming the SV-like condition for devices, we show that the honest parties can achieve amplification of the weak source, for the parameter range 0 ≤ ε <; 0.0132, against a class of attacks given as a mixture of product box sequences, made of extremal no-signaling boxes, with additional symmetry conditions. Composable security proof against this class of attacks is provided. Show less

Recently, the physically realistic protocol amplifying the randomness of Santha-Vazirani sources producing cryptographically secure random bits was proposed; however, for reasons of practical relevance, the crucial question remained open regarding whether this can be accomplished under the minimal conditions necessary for the task. Namely, is it possible to achieve randomness amplification using only two no-signaling components and in a situation where the violation of a Bell inequality only…

Recently, the physically realistic protocol amplifying the randomness of Santha-Vazirani sources producing cryptographically secure random bits was proposed; however, for reasons of practical relevance, the crucial question remained open regarding whether this can be accomplished under the minimal conditions necessary for the task. Namely, is it possible to achieve randomness amplification using only two no-signaling components and in a situation where the violation of a Bell inequality only guarantees that some outcomes of the device for specific inputs exhibit randomness? Here, we solve this question and present a device-independent protocol for randomness amplification of Santha-Vazirani sources using a device consisting of two nonsignaling components. We show that the protocol can amplify any such source that is not fully deterministic into a fully random source while tolerating a constant noise rate and prove the composable security of the protocol against general no-signaling adversaries. Show less

Randomness is a fundamental concept, with implications from security of modern data systems, to fundamental laws of nature and even the philosophy of science. Randomness is called certified if it describes events that cannot be pre-determined by an external adversary. It is known that weak certified randomness can be amplified to nearly ideal randomness using quantum-mechanical systems. However, so far, it was unclear whether randomness amplification is a realistic task, as the existing…

Randomness is a fundamental concept, with implications from security of modern data systems, to fundamental laws of nature and even the philosophy of science. Randomness is called certified if it describes events that cannot be pre-determined by an external adversary. It is known that weak certified randomness can be amplified to nearly ideal randomness using quantum-mechanical systems. However, so far, it was unclear whether randomness amplification is a realistic task, as the existing proposals either do not tolerate noise or require an unbounded number of different devices. Here we provide an error-tolerant protocol using a finite number of devices for amplifying arbitrary weak randomness into nearly perfect random bits, which are secure against a no-signalling adversary. The correctness of the protocol is assessed by violating a Bell inequality, with the degree of violation determining the noise tolerance threshold. An experimental realization of the protocol is within reach of current technology. Show less