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Assuming that a threshold Ornstein-Uhlenbeck process is observed at discrete time instants, we propose generalized moment estimators to estimate the parameters. Our theoretical basis is the celebrated ergodic theorem. To use this theorem we…
We demonstrate that any Euclidean-time quantum mechanical theory may be represented as a neural network, ensured by the Kosambi-Karhunen-Lo\`eve theorem, mean-square path continuity, and finite two-point functions. The additional constraint…
Mostof the existing literature on supervised machine learning problems focuses on the case when the training data set is drawn from an i.i.d. sample. However, many practical problems are characterized by temporal dependence and strong…
We obtain complementary recurrence and transience criteria for processes $X=(X_n)_{n \ge 0}$ with values in $\mathbb R^d_+$ fulfilling a non-linear equation $X_{n+1}=MX_n+g(X_n)+ \xi_{n+1}$. Here $M$ denotes a primitive matrix having…
We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the…
We study the ergodic properties of eigenfunctions of Schr\"odinger operators on a closed connected Riemannian manifold $M$ in case that the underlying Hamiltonian system possesses certain symmetries. More precisely, let $M$ carry an…
We give an equivalent finitary reformulation of the classical Shannon-McMillan-Breiman theorem which has an immediate translation to the case of ergodic quantum lattice systems. This version of a quantum Breiman theorem can be derived from…
We introduce a class of discrete random walk model driven by global memory effects. At any time the right-left transitions depend on the whole previous history of the walker, being defined by an urn-like memory mechanism. The characteristic…
In this study, a new extension of the Markov Renewal theory is introduced by allowing time to evolve in multiple dimensions. The resulting chains are referred to as multi-time Markov Renewal chains and since this extension is new, the state…
Differential geometry offers a powerful framework for optimising and characterising finite-time thermodynamic processes, both classical and quantum. Here, we start by a pedagogical introduction to the notion of thermodynamic length. We…
It is a well known general principle that the Fourier transform of a random measure is small, except at the zero frequency, in various senses for appropriate notions of randomness. In this note we develop analogues of this principle for two…
We establish a new class of functional central limit theorems for partial sum of certain symmetric stationary infinitely divisible processes with regularly varying L\'{e}vy measures. The limit process is a new class of symmetric stable…
This article traces the development of fluctuation theory and its deep connection to irreversibility, from equilibrium to near-equilibrium, and finally to far-from-equilibrium systems. Classical fluctuation theorems, which capture the…
We describe some basic results for Quantum Stochastic Processes and present some new results about a certain class of processes which are associated to Quantum Iterated Function Systems (QIFS). We discuss questions related to the Markov…
We focus on a data sequence produced by repetitive quantum measurement on an internal hidden quantum system, and call it a hidden Markovian process. Using a quantum version of the Perron-Frobenius theorem, we derive novel upper and lower…
In this short note we prove ``effective" geometric ergodicity (i.e a Perron-Frobenius theorem) for Markov chains in random mixing dynamical environment satisfying a random non-uniform version of the Doeblin condition. Effectivity here means…
We argue that quantum theory is a low-energy effective theory which emerges from some sub-quantum level theory which is of an undulatory and translocal character. We show the close connection of quantum theory with both gravity and the…
We study ergodic properties of certain piecewise smooth two-dimensional systems by constructing countable Markov partitions. Using thermodynamic formalism we prove exponential decay of correleations.
We prove that the stationarity and the ergodicity of a quantum source are preserved by any trace-preserving completely positive linear map of the tensor product form ${\cal E}^{\otimes m}$, where a copy of ${\cal E}$ acts locally on each…
Local mean and individual (with respect to almost uniform convergence in Egorov's sense) ergodic theorems are established for actions of the semigroup $\mathbb R_+^d$ in symmetric spaces of measurable operators associated with a semifinite…