Generalized heredity in ℬ-free systems

Let [Formula: see text] be a primitive set, [Formula: see text], [Formula: see text], and denote by [Formula: see text] the orbit closure of [Formula: see text] under the shift. We complement results on heredity of [Formula: see text] from [Dymek et al., [Formula: see text]-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425–5489] in two directions: In the proximal case we prove that a certain subshift [Formula: see text], which coincides with [Formula: see text] when [Formula: see text] is taut, is always hereditary. (In particular there is no need for the stronger assumption that the set [Formula: see text] has light tails, as in [Dymek et al., [Formula: see text]-free sets and dynamics, Trans. Amer. Math. Soc. 370 (2018) 5425–5489].) We also generalize the concept of heredity to include the non-proximal (and hence non-hereditary) case by proving that [Formula: see text] is always “hereditary above its unique minimal (Toeplitz) subsystem”. Finally, we characterize this Toeplitz subsystem as being a set [Formula: see text], where [Formula: see text] for a set [Formula: see text] that can be derived from [Formula: see text], and draw some further conclusions from this characterization. Throughout results from [Kasjan et al., Dynamics of [Formula: see text]-free sets: A view through the window, Int. Math. Res. Not. 2019 (2019) 2690–2734] are heavily used.

On perpetuities with light tails

In this paper we consider the asymptotics of logarithmic tails of a perpetuity R=D∑j=1∞Qj∏k=1j-1Mk, where (Mn,Qn)n=1∞ are independent and identically distributed copies of (M,Q), for the case when ℙ(M∈[0,1))=1 and Q has all exponential moments. If M and Q are independent, under regular variation assumptions, we find the precise asymptotics of -logℙ(R>x) as x→∞. Moreover, we deal with the case of dependent M and Q, and give asymptotic bounds for -logℙ(R>x). It turns out that the dependence structure between M and Q has a significant impact on the asymptotic rate of logarithmic tails of R. Such a phenomenon is not observed in the case of heavy-tailed perpetuities.

Calculating State-Dependent Noise in a Linear Inverse Model Framework

The most commonly used version of a linear inverse model (LIM) is forced by state-independent noise. Although having several desirable qualities, this formulation can only generate long-term Gaussian statistics. LIM-like systems forced by correlated additive–multiplicative (CAM) noise have been shown to generate deviations from Gaussianity, but parameter estimation methods are only known in the univariate case, limiting their use for the study of coupled variability. This paper presents a methodology to calculate the parameters of the simplest multivariate LIM extension that can generate long-term deviations from Gaussianity. This model (CAM-LIM) consists of a linear deterministic part forced by a diagonal CAM noise formulation, plus an independent additive noise term. This allows for the possibility of representing asymmetric distributions with heavier- or lighter-than-Gaussian tails. The usefulness of this methodology is illustrated in a locally coupled two-variable ocean–atmosphere model of midlatitude variability. Here, a CAM-LIM is calculated from ocean weather station data. Although the time-resolved dynamics is very close to linear at a time scale of a couple of days, significant deviations from Gaussianity are found. In particular, individual probability density functions are skewed with both heavy and light tails. It is shown that these deviations from Gaussianity are well accounted for by the CAM-LIM formulation, without invoking nonlinearity in the time-resolved operator. Estimation methods using knowledge of the CAM-LIM statistical constraints provide robust estimation of the parameters with data lengths typical of geophysical time series, for example, 31 winters for the ocean weather station here.

Validation of aggregated risks models

Validation of risk models is required by regulators and demanded by management and shareholders. Those models rely in practice heavily on Monte Carlo (MC) simulations. Given their complexity, the convergence of the MC algorithm is difficult to prove mathematically. To circumvent this problem and nevertheless explore the conditions of convergence, we suggest an analytical approach. Considering standard models, we compute, via mixing techniques, closed form formulas for risk measures as Value-at-Risk (VaR) VaR or Tail Value-at-Risk (TVaR) TVaR on a portfolio of risks, and consequently for the associated diversification benefit. The numerical convergence of MC simulations of those various quantities is then tested against their analytical evaluations. The speed of convergence appears to depend on the fatness of the tail of the marginal distributions; the higher the tail index, the faster the convergence. We also explore the behaviour of the diversification benefit with various dependence structures and marginals (heavy and light tails). As expected, it varies heavily with the type of dependence between aggregated risks. The diversification benefit is also studied as a function of the risk measure, VaR or TVaR.

On designing an active tail for legged robots: simplifying control via decoupling of control objectives

Purpose The purpose of this paper is to explore the possible roles of active tails for steady-state legged locomotion, focusing on a design principle which simplifies control by decoupling different control objectives. Design/methodology/approach A series of simple models are proposed which capture the dynamics of an idealized running system with an active tail. These models suggest that the overall control problem can be simplified and effectively decoupled via a proper tail design. This design principle is further explored in simulation using trajectory optimization. The results are then validated in hardware using a one degree-of-freedom active tail mounted on the quadruped robot Cheetah-Cub. Findings The results of this paper show that an active tail can greatly improve both forward velocity and reduce body-pitch per stride while adding minimal complexity. Further, the results validate the design principle of using long, light tails compared to shorter heavier ones. Originality/value This paper builds on previous results, with a new focus on steady-state locomotion and in particular deals directly with stance phase dynamics. A novel design principle for tails is proposed and validated.

Large deviations for multidimensional state-dependent shot-noise processes

Shot-noise processes are used in applied probability to model a variety of physical systems in, for example, teletraffic theory, insurance and risk theory, and in the engineering sciences. In this paper we prove a large deviation principle for the sample-paths of a general class of multidimensional state-dependent Poisson shot-noise processes. The result covers previously known large deviation results for one-dimensional state-independent shot-noise processes with light tails. We use the weak convergence approach to large deviations, which reduces the proof to establishing the appropriate convergence of certain controlled versions of the original processes together with relevant results on existence and uniqueness.

Large deviations for multidimensional state-dependent shot-noise processes

Shot-noise processes are used in applied probability to model a variety of physical systems in, for example, teletraffic theory, insurance and risk theory, and in the engineering sciences. In this paper we prove a large deviation principle for the sample-paths of a general class of multidimensional state-dependent Poisson shot-noise processes. The result covers previously known large deviation results for one-dimensional state-independent shot-noise processes with light tails. We use the weak convergence approach to large deviations, which reduces the proof to establishing the appropriate convergence of certain controlled versions of the original processes together with relevant results on existence and uniqueness.