The noise in control systems was studied based on data from several hundreds of control loops operating in different process industries located in several sites all over the world. That data showed that the theoretical assumption of Gaussian properties for the data is hardly ever satisfied. This paper will focus on some illustrative examples of stochastic models with non-Gaussian noise and will present the evolution process in using stochastic processes that include fractional Brownian motion processes, Rosenblatt and Rosenblatt–Volterra processes as a replacement of commonly used ordinary Brownian motions. Theoretical advancements will demonstrate challenges and fascinating opportunities in them for developing the models that meet the expectations of industrial practitioners.

A scalar linear stochastic system with a Rosenblatt noise process is described and a prediction problem is explicitly solved for this system. Some other topics related to this stochastic system are also considered. Since the Rosenblatt processes are defined by double Wiener-Itô integrals the usual stochastic calculus for Brownian motion is not appropriate. Stochastic calculus is not used for the double integrals. Instead one uses the Wiener-ô double integral definition that has an appropriate orthogonality for expectation. For the prediction problem solution instead one uses some stochastic calculus for a double Wiener-Itô so that such integrals have expectation zero. An ergodic control for a scalar controlled system with a Rosenblatt noise is also briefly described.

This paper examines a class of mean-field-type games with finite numbers of decision-makers with state dynamics driven by Rosenblatt processes. Rosen-blatt processes are non-Gaussian, non-Poisson, and non-Markov with long-range dependence. We provide equilibrium strategies and equilibrium costs in linear state-and-mean-field-type feedback form for all decision-makers. Interestingly, we show that the equilibrium strategies for state driven by Brownian, multi-fractional Brownian, Gauss-Volterra processes no longer provide equilibrium in presence of Rosenblatt processes.

In this paper, we propose a co-opetitive mean-field-type game (MFTG) approach in which decision makers interact with each other by means of partial cooperation and competition simultaneously. The proposed novel approach allows decision makers' preferences to evolve over time to cooperate with those who contribute to their utilities, and to compete with those who are working against their individual interests. In addition, we consider that each decision maker has a co-opetitive capacity/power. The co-opetitive MFTG involves concepts, such as selfishness, altruism, competition, cooperation, among others; all together within the same strategic interaction. Both co-opetitive MFTGs and evolutionary dynamics describing the evolution of co-opetitive parameters are combined. We design incentives to promote the emergence of co-opetition over selfish behavior.

Context trees of arbitrary stationary ergodic processes with finite alphabets are considered. Such a process is not necessarily a Markov chain, so the context tree may be of infinite depth. Calculated from a sample of sizen, the Bayesian information criterion (BIC) is shown to provide a strongly consistent estimator of the context tree of the process, via minimization over hypothetical context trees, without any restriction on the hypothetical context trees. Strong consistency means that the estimated context tree recovers the true one up to a levelK, eventually almost surely asntends to infinity. Under some conditions on the process, it is shown that the recovery levelKcan grow withnat a specific rate determined by the distribution of the process; thus, the BIC estimator can recover the true context tree to larger and larger depths. The results include for the special case ofKbeing an arbitrary constant that the strong consistency is satisfied without any assumption on the stationary ergodic process, which itself improves the existing results, where either the true context tree was assumed to be of finite depth or the depth of the hypothetical context trees was bounded byo(logn).

In this paper, some relations between estimation and mutual information are given by expressing two mutual information calculations in terms of two distinct estimation errors. Specifically the mutual information between a stochastic signal and a pure jump Levy process whose rate function depends on the signal is expressed in terms of a filtering error and the rate of change of this mutual information with respect to a parameter multiplying the rate function of the Levy process is expressed in terms of a smoothing error. These results generalize the analogous mutual information results for some Gaussian noise processes with additive stochastic signals.

Context trees of arbitrary stationary ergodic processes with finite alphabets are considered. Such a process is not necessarily a Markov chain, so the context tree may be of infinite depth. Calculated from a sample of size n, the Bayesian information criterion (BIC) is shown to provide a strongly consistent estimator of the context tree of the process, via minimization over hypothetical context trees, without any restriction on the hypothetical context trees. Strong consistency means that the estimated context tree recovers the true one up to any fixed level K, eventually almost surely as n tends to infinity. This generalizes the previous results, where either the context trees were assumed to be of finite depth or the depth of the hypothetical context trees was bounded by O(log n). Moreover, under some conditions on the process it is also shown that the level K above can grow with n at a specific rate determined by the distribution of the process; thus the BIC estimator can recover the true context tree to larger and larger depths.

The mutual information between a stochastic signal and this signal plus a fractional Brownian motion (described as an additive fractional Gaussian noise channel) is expressed as the error of an estimation problem that can be naturally associated with this model. If the stochastic signal with the additive fractional Brownian motion occurs multiplied by a scalar parameter, then the rate of change of the mutual information with respect to this parameter is described by the error of another related estimation problem. These results generalize some results for a model where the fractional Brownian motion is a Brownian motion to a model with an arbitrary fractional Brownian motion.