In a previous paper of Winter and the author the Law of Large Numbers for the local mass of certain superdiffusions was proved under a spectral theoretical assumption, which is equivalent to the ergodicity (positive recurrence) of the motion component of an $H$-transformed (or weighted) superprocess. In fact the assumption is also equivalent to the property that the scaling for the expectation of the local mass is pure exponential.
In this paper we go beyond ergodicity, that is we consider cases when the scaling is not purely exponential. Inter alia, we prove the analog of the Watanabe-Biggins Law of Large Numbers for super-Brownian motion (SBM).
We will also prove another Law of Large Numbers for a bounded set moving with subcritical speed, provided the variance term decays sufficiently fast.
In this paper we establish a weak and a strong law of large numbers for supercritical superprocesses with general non-local branching mechanisms. Our results complement earlier results obtained for superprocesses with only local branching. Several interesting examples are developed, including multitype continuous-state branching processes, multitype superdiffusions and superprocesses with discontinuous spatial motions and non-decomposable branching mechanisms.
We study the empirical spectral distribution (ESD) of symmetric random matrices with ergodic entries on the diagonals. We observe that for entries with correlations that decay to 0, when the distance of the diagonal entries becomes large the limiting ESD is the well known semicircle law. If it does not decay to 0 (and have the same sign) the semicircle law cannot be the limit of the ESD. This is good agreement with results on exchangeable processes analysed in Friesen and Löwe (2013a) and Hochstättler et al. (2016).
We are interested in the dynamic of a structured branching population where the trait of each individual moves according to a Markov process. The rate of division of each individual is a function of its trait and when a branching event occurs, the trait of a descendant at birth depends on the trait of the mother. We prove a law of large numbers for the empirical distribution of ancestral trajectories. It ensures that the empirical measure converges to the mean value of the spine which is a time-inhomogeneous Markov process describing the trait of a typical individual along its ancestral lineag…
We consider noise perturbations of delay differential equations (DDE) experiencing Hopf bifurcation. The noise is assumed to be exponentially ergodic, i.e. transition density converges to stationary density exponentially fast uniformly in the initial condition. We show that, under an appropriate change of time scale, as the strength of the perturbations decreases to zero, the law of the critical eigenmodes converges to the law of a diffusion process (without delay). We prove the result only for scalar DDE. For vector-valued DDE without proofs see Phys.Rev.E v93, 062104.
A more general notion of weight called admissible is introduced and then an investigation is carried out on the a.e. convergence of weighted strong laws of large numbers and their applications to weighted one-sided ergodic Hilbert transforms on vector-valued $L_p$-spaces. Furthermore, the obtained results are applied to the a.e. convergence of random modulated weighted ergodic series, which is also new in the classical setting.
We study the ergodic behaviour of a discrete-time process $X$ which is a Markov chain in a stationary random environment. The laws of $X_t$ are shown to converge to a limiting law in (weighted) total variation distance as $t\to\infty$. Convergence speed is estimated and an ergodic theorem is established for functionals of $X$. Our hypotheses on $X$ combine the standard "small set" and "drift" conditions for geometrically ergodic Markov chains with conditions on the growth rate of a certain "maximal process" of the random environment. We are able to cover a wide range of models that have hereto…
In previous work by Avena and den Hollander, a model of a one-dimensional random walk in a dynamic random environment was proposed where the random environment is resampled from a given law along a growing sequence of deterministic times. In the regime where the increments of the resampling times diverge, which is referred to as the cooling regime, a weak law of large numbers and certain fluctuation properties were derived under the annealed measure. In the present paper we show that a strong law of large numbers and a quenched large deviation principle hold as well. In the cooling regime, the…
Long-time dynamics of stochastic wave equation with dissipative damping and its full discretization: exponential ergodicity and strong law of large numbers
This brief pedagogical note re-proves a simple theorem on the convergence, in $L_2$ and in probability, of time averages of non-stationary time series to the mean of expectation values. The basic condition is that the sum of covariances grows sub-quadratically with the length of the time series. I make no claim to originality; the result is widely, but unevenly, spread bit of folklore among users of applied probability. The goal of this note is merely to even out that distribution.
We study a G/GI/1 single-server queuing model with i.i.d.\ service times that are independent of a stationary process of inter-arrival times. We show that the distribution of the waiting time converges to a stationary law as time tends to infinity provided that inter-arrival times satisfy a Gärtner-Ellis type condition. A convergence rate is given and a law of large numbers established. These results provide tools for the statistical analysis of such systems, transcending the standard case with independent inter-arrival times.
In this expository note, we give a simple proof that a gambler repeating a game with positive expected value never goes broke with a positive probability. This does not immediately follow from the strong law of large numbers or other basic facts on random walks. Using this result, we provide an elementary proof of the strong law of large numbers. The ideas of the proofs come from the maximal ergodic theorem and Birkhoff's ergodic theorem.
We study a family of McKean-Vlasov (mean-field) type ergodic optimal control problems with linear control, and quadratic dependence on control of the cost function. For this class of problems we establish existence and uniqueness of an optimal control. We propose an $N$-particles Markovian optimal control problem approximating the McKean-Vlasov one and we prove the convergence in relative entropy, total variation and Wasserstein distance of the law of the former to the law of the latter when $N$ goes to infinity. Some McKean-Vlasov optimal control problems with singular cost function and the r…
The ergodic theorems of Hopf, Wiener and Birkhoff were extended to the context of Riesz spaces with a weak order unit and conditional expectation operator by Kuo, Labuschagne and Watson in [Ergodic Theory and the Strong Law of Large Numbers on Riesz Spaces. Journal of Mathematical Analysis and Applications, 325,(2007), 422-437.]. However, the precise concept of what constitutes ergodicity in Riesz spaces was not considered. In this short paper we fill in this omission and give some explanations of the choices made. In addition, we consider the interplay between mixing and ergodicity in the Rie…
