Growth–fragmentation–coagulation equations with unbounded coagulation kernels

In this paper, we prove the global in time solvability of the continuous growth–fragmentation–coagulation equation with unbounded coagulation kernels, in spaces of functions having finite moments of sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth–fragmentation semigroup that allows us to consider coagulation kernels whose growth for large clusters is controlled by how good the regularization is, in a similar manner to the case when the semigroup is analytic. This article is part of the theme issue ‘Semigroup applications everywhere’.

Functional ARCH and GARCH Models: A Yule-Walker Approach

Conditional heteroskedastic financial time series are commonly modelled by (G)ARCH processes. ARCH(1) and GARCH were recently established in C[0,1] and L^2[0,1]. This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of (G)ARCH processes for any order in C[0,1] and L^p[0,1]. It deduces explicit asymptotic upper bounds of estimation errors for the shift term, the complete (G)ARCH operators and the projections of ARCH operators on finite-dimensional subspaces. The operator estimaton is based on Yule-Walker equations, and estimating the GARCH operators also involves a result estimating operators in invertible linear processes being valid beyond the scope of (G)ARCH. Moreover, our results regarding (G)ARCH can be transferred to functional AR(MA).

Dynamics of two qubits in common environment

We consider the entanglement evolution of two qubits embedded into disordered multiconnected environment. We model the environment and its interaction with qubits by large random matrices allowing for a possibility to describe environments of meso- and even nanosize. We obtain general formulas for the time dependent reduced density matrix of the qubits corresponding to several cases of the qubit-environment interaction and initial condition. We then work out an analog of the Born–Markov approximation to find the evolution of the widely used entanglement quantifiers: the concurrence, the negativity and the quantum discord. We show that even in this approximation the time evolution of the reduced density matrix can be non-Markovian, thereby describing certain memory effects due to the backaction of the environment on qubits. In particular, we find the vanishing of the entanglement (Entanglement Sudden Death) at finite moments and its revivals (Entanglement Sudden Birth). Our results, partly known and partly new, can be viewed as a manifestation of the universality of certain properties of decoherent qubit evolution which have been found previously in various versions of bosonic macroscopic environment.

Robust Test for Detecting Changes in the Autocovariance Function of a Time Series

We propose a new robust test to detect changes in the autocovariance function of a time series. The test is based on empirical autocovariances of a robust transformation of the original time series. Because of the transformation, we do not require any finite moments of the original time series, making the test especially suitable for heavy tailed time series. We furthermore propose a lag weighting scheme, which puts emphasis on changes of the autocovariance at smaller lags. Our approach is compared to existing ones in some simulations.

Uncorrelatedness sets of discrete random variables via Vandermonde-type determinants

Given random variables X and Y having finite moments of all orders, their uncorrelatedness set is defined as the set of all pairs (j, k) ∈ ℕ2, for which Xj and Yk are uncorrelated. It is known that, broadly put, any subset of ℕ2 can serve as an uncorrelatedness set. This claim is no longer valid for random variables with prescribed distributions, in which case the need arises so as to identify the possible uncorrelatedness sets. This paper studies the uncorrelatedness sets for positive random variables uniformly distributed on three points. Some general features of these sets are derived. Two related Vandermonde-type determinants are examined and applied to describe uncorrelatedness sets in some special cases.

Lyapunov Spectrum of Nonautonomous Linear Young Differential Equations

We show that a linear Young differential equation generates a topological two-parameter flow, thus the notions of Lyapunov exponents and Lyapunov spectrum are well-defined. The spectrum can be computed using the discretized flow and is independent of the driving path for triangular systems which are regular in the sense of Lyapunov. In the stochastic setting, the system generates a stochastic two-parameter flow which satisfies the integrability condition, hence the Lyapunov exponents are random variables of finite moments. Finally, we prove a Millionshchikov theorem stating that almost all, in a sense of an invariant measure, linear nonautonomous Young differential equations are Lyapunov regular.

A characterization of probability measures in terms of semi-quantum operators

A canonical definition of the joint semi-quantum operators of a finite family of random variables, having finite moments of all orders, is given first in terms of an existence and uniqueness theorem. Then two characterizations, one for the polynomially symmetric, and another for the polynomially factorizable probability measures, having finite moments of all orders, are presented.

Sequential probability ratio test for many simple hypotheses on parameters of time series with trend

The problem of sequential test for many simple hypotheses on parameters of time series with trend is considered. Two approaches, including M-ary sequential probability ratio test and matrix sequential probability ratio test are used for constructing the sequential test. The sufficient conditions of finite terminations of the test and the existence of finite moments of their stopping times are given. The upper bounds for the average numbers of observations are obtained. With the thresholds chosen suitably, these tests can belong to some specified classes of statistical tests. Numerical examples are presented.

The Asymptotic Behavior of Particle Size Distribution Undergoing Brownian Coagulation Based on the Spline-Based Method and TEMOM Model

In this paper, the particle size distribution is reconstructed using finite moments based on a converted spline-based method, in which the number of linear system of equations to be solved reduced from 4m × 4m to (m + 3) × (m + 3) for (m + 1) nodes by using cubic spline compared to the original method. The results are verified by comparing with the reference firstly. Then coupling with the Taylor-series expansion moment method, the evolution of particle size distribution undergoing Brownian coagulation and its asymptotic behavior are investigated.