This paper presents the first generalization bounds for time series prediction with a non-stationary mixing stochastic process. We prove Rademacher complexity learning bounds for both average-path generalization with non-stationary β-mixing processes and path-dependent generalization with non-stationary φ-mixing processes. Our guarantees are expressed in terms of βor φ-mixing coefficients and a...

Assume that a stochastic process can be approximated, when some scale parameter gets large, by a fluid limit (also called “mean field limit”, or “hydrodynamic limit”). A common practice, often called the “fixed point approximation” consists in approximating the stationary behaviour of the stochastic process by the stationary points of the fluid limit. It is known that this may be incorrect in g...

We investigate in this paper the stability of non-stationary stochastic processes, arising typically in applications of control. The setting is known as stochastic recursive sequences, which allows us to construct on one probability space stochastic processes that correspond to diierent initial states and even diierent control policies. It does not require any Markovian assumptions. A natural c...

The existence of stochastic processes is discussed, describing turbulent solutions of the full Navier-Stokes equation, driven by unidirectional flow, in dimensions one, two and three. These solutions turn out to have a finite velocity and velocity gradient but they are not smooth instead the velocity is Hölder continuous with a Hölder exponent depending on the dimension. They scale with the Kol...

We continue the study of a model for heat conduction [6] consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the infinit...

We present a generalization of Szegö theory of orthogonal polynomials on the unit circle to orthogonal rational functions. Unlike previous results, the poles of the rational functions may tend to the unit circle under smoothness assumptions on the density of the measure. Just like the Kolmogorov-KreinSzegö theorem may be interpreted as an asymptotic estimate of the prediction error for stationa...

Vector-valued, asymptotically stationary stochastic processes on r-compact locally compact abelian groups are studied. For such processes, we introduce a stationary spectral measure and show that it is discrete if and only if the asymptotically stationary covariance function is almost periodic. Using an “almost periodic Fourier transform” we recover the discrete part of the spectral measure and...

We obtain an invariance principle for non-stationary vector-valued stochastic processes. It is shown that, under mild conditions, the partial sums of non-stationary processes can be approximated on a richer probability space by sums of independent Gaussian random vectors with nearly optimal bounds. The latter Gaussian approximation result has a wide range of applications in the study of multipl...

This paper generalizes the notion of stochastic order to a relation between probability measures over arbitrary measurable spaces. This generalization is motivated by the observation that for the stochastic ordering of two stationary Markov processes, it suffices that the generators of the processes preserve some, not necessarily reflexive or transitive, subrelation of the order relation. The m...