A Lepskiĭ-type stopping rule for the covariance estimation of multi-dimensional Lévy processes

We suppose that a Lévy process is observed at discrete time points. Starting from an asymptotically minimax family of estimators for the continuous part of the Lévy Khinchine characteristics, i.e., the covariance, we derive a data-driven parameter choice for the frequency of estimating the covariance. We investigate a Lepskiĭ-type stopping rule for the adaptive procedure. Consequently, we use a balancing principle for the best possible data-driven parameter. The adaptive estimator achieves almost the optimal rate. Numerical experiments with the proposed selection rule are also presented.

Bayesian analysis of two-part nonlinear latent variable model: Semiparametric method

Two-part model (TPM) is a widely appreciated statistical method for analyzing semi-continuous data. Semi-continuous data can be viewed as arising from two distinct stochastic processes: one governs the occurrence or binary part of data and the other determines the intensity or continuous part. In the regression setting with the semi-continuous outcome as functions of covariates, the binary part is commonly modelled via logistic regression and the continuous component via a log-normal model. The conventional TPM, still imposes assumptions such as log-normal distribution of the continuous part, with no unobserved heterogeneity among the response, and no collinearity among covariates, which are quite often unrealistic in practical applications. In this article, we develop a two-part nonlinear latent variable model (TPNLVM) with mixed multiple semi-continuous and continuous variables. The semi-continuous variables are treated as indicators of the latent factor analysis along with other manifest variables. This reduces the dimensionality of the regression model and alleviates the potential multicollinearity problems. Our TPNLVM can accommodate the nonlinear relationships among latent variables extracted from the factor analysis. To downweight the influence of distribution deviations and extreme observations, we develop a Bayesian semiparametric analysis procedure. The conventional parametric assumptions on the related distributions are relaxed and the Dirichlet process (DP) prior is used to improve model fitting. By taking advantage of the discreteness of DP, our method is effective in capturing the heterogeneity underlying population. Within the Bayesian paradigm, posterior inferences including parameters estimates and model assessment are carried out through Markov Chains Monte Carlo (MCMC) sampling method. To facilitate posterior sampling, we adapt the Polya-Gamma stochastic representation for the logistic model. Using simulation studies, we examine properties and merits of our proposed methods and illustrate our approach by evaluating the effect of treatment on cocaine use and examining whether the treatment effect is moderated by psychiatric problems.

Mixed Spectra for Stable Signals from Discrete Observations

This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.

Aliasing Free For Mixed Spectra for Stable Processes

This work focuses on the symmetric alpha stable processes with continuous time frequently used in modeling the signal with indefinitely growing variance when the spectral measure is mixed: sum of a continuous meseare and discrete measure. The objective of this paper is to estimate the spectral density of the continuous part from discrete observations of the signal. For that, we propose a method based on a sample of the signal at a periodic instant. The Jackson polynomial kernel is used for construct a periodogram. We smooth this periodogram by two spectral windows taking into account the width of the interval where the spectral density is nonzero. This technique allows to circumvent the phenomenon of aliasing often encountered in the estimation from the discrete observations of a process with a continuous time.

Distributions for Nonsymmetric Monotone and Weakly Monotone Position Operators

We study the vacuum distribution, under an appropriate scaling, of a family of partial sums of nonsymmetric position operators on weakly monotone and monotone Fock spaces, respectively. We preliminary treat the case of weakly monotone Fock space, and show that any single operator has the vacuum law belonging to the free Meixner class. After establishing some relations between the combinatorics of Motzkin and Riordan paths, we give a recursive formula for the vacuum moments of the law of any finite sum. Since the operators are monotone independent, the distribution is the monotone convolution of the free Meixner law above. We also investigate the asymptotic measure for these sums, which can be seen as “Poisson type” limit law. It turns out to belong to the free Meixner class, with an atomic and an absolutely continuous part (w.r.t. the Lebesgue measure). Finally, we briefly apply analogous considerations to the case of monotone Fock space.

Memory Effects in Quantum Dynamics Modelled by Quantum Renewal Processes

Simple, controllable models play an important role in learning how to manipulate and control quantum resources. We focus here on quantum non-Markovianity and model the evolution of open quantum systems by quantum renewal processes. This class of quantum dynamics provides us with a phenomenological approach to characterise dynamics with a variety of non-Markovian behaviours, here described in terms of the trace distance between two reduced states. By adopting a trajectory picture for the open quantum system evolution, we analyse how non-Markovianity is influenced by the constituents defining the quantum renewal process, namely the time-continuous part of the dynamics, the type of jumps and the waiting time distributions. We focus not only on the mere value of the non-Markovianity measure, but also on how different features of the trace distance evolution are altered, including times and number of revivals.

Compacting oblivious agents on dynamic rings

In this paper we investigate dynamic networks populated by autonomous mobile agents. Dynamic networks are networks whose topology can change continuously, at unpredictable locations and at unpredictable times. These changes are not considered to be faults, but rather an integral part of the nature of the system. The agents can autonomously move on the network, with the goal of solving cooperatively an assigned common task. Here, we focus on a specific network: the unoriented ring. More specifically, we study 1-interval connected dynamic rings (i.e., at any time, at most one of the edges might be missing). The agents move according to the widely used Look–Compute–Move life cycle, and can be homogenous (thus identical) or heterogenous (agents are assigned colors from a set of c > 1 colors). For identical agents, their goal is to form a compact segment, where agents occupy a continuous part of the ring and no two agents occupy the same node: we call this the Compact Configuration Problem. In the case of agents with colors, called the Colored Compact Configuration Problem, the goal is to group agents such that each group is formed by all agents having the same color, it occupies a continuous segment of the network, and groups of agents having different colors occupy distinct areas of the network. In this paper we determine the necessary conditions to solve both proposed problems. For all solvable cases, we provide algorithms for both the monochromatic and the colored version of the compact configuration problem. All our algorithms work even for the simplest model where agents have no persistent memory, no communication capabilities and do not agree on a common orientation within the network. To the best of our knowledge this is the first work on the compaction problem in a dynamic network.

Control of negative gain nonlinear processes using sliding mode controllers with modified Nelder-Mead tuning equations

This paper proposes a sliding mode controller (SMC) with modified Nelder-Mead tuning, for the control of nonlinear chemical processes, which are represented as first order plus dead time process with negative gain (FOPDT-NG). In the controller design, the SMC controller parameter in continuous part is obtained based on the time constant and dead time of the process, and controller parameters in the discontinuous part is obtained using Nelder-Mead tuning equations. Even though the controller parameters of conventional SMC are tuned using Nelder-Mead tuning, zero dynamics are noticed in the closed loop response of few FOPDT-NG processes and, with few other FOPDT-NG processes tracking of set-point is unachievable. This work proposes modification in the Nelder-Mead tuning equations using Nelder-Mead optimization to overcome the above disadvantages. Four different types of FOPDT-NG processes are considered in this work, and for every type the Nelder-Mead tuning equations are modified, for the design of proposed controllers. The performances of proposed controllers are evaluated for FOPDT-NG processes and also for three different chemical processes taken from literature. A simulation results demonstrate that, the proposed controller prevailed the performance of the conventional SMC in tracking the set-point and the elimination of zero dynamic behavior of FOPDT-NG processes. Hence, the proposed controllers provide improved closed loop performances as compared to the conventional SMC.

Regularity Properties of Free Multiplicative Convolution on the Positive Line

Given two nondegenerate Borel probability measures $\mu$ and $\nu$ on ${\mathbb{R}}_{+}=[0,\infty )$, we prove that their free multiplicative convolution $\mu \boxtimes \nu$ has zero singular continuous part and its absolutely continuous part has a density bounded by $x^{-1}$. When $\mu$ and $\nu$ are compactly supported Jacobi measures on $(0,\infty )$ having power law behavior with exponents in $(-1,1)$, we prove that $\mu \boxtimes \nu$ is another Jacobi measure whose density has square root decay at the edges of its support.

The ruin problem for a Wiener process with state-dependent jumps

Let X(t) be a jump-diffusion process whose continuous part is a Wiener process, and let T (x) be the first time it leaves the interval (0,b), where x = X(0). The jumps are negative and their sizes depend on the value of X(t). Moreover there can be a jump from X(t) to 0. We transform the integro-differential equation satisfied by the probability p(x) := P[X(T (x)) = 0] into an ordinary differential equation and we solve this equation explicitly in particular cases. We are also interested in the moment-generating function of T (x).