We introduce a notion called “intrinsic location functional”. This is a large family of random locations including, for example, the location of the path supremum/infimum over an interval, the first/last hitting time, among many others. On one hand, it is proved that under stationarity, the distributions of intrinsic location functionals must satisfy the same very specific constraints, in spite...

Starting from known necessary extremality conditions in terms of strict subdifferentials and normals the notion of weak stationarity is introduced. It is defined in terms of initial space elements. The necessary conditions become necessary and sufficient (for stationarity).

EXt = μ Cov(Xt, Xt−k) = γk (lag-k autocovariance). The lag zero autocovariance γ0 is just the variance of the time series. γk’s are of central importance in time series analysis as they characterize the serial dependence over observations (i.e. over time). We usually do not have iid data in time series contexts. A white noise series is stationary. A white noise (WN) series is defined as an unco...

We present an axiom system for a weak form of expectation operators leading to an interpretation in which a probabilistic process is modelled as a set of Markov matrices rather than a single matrix. Such`generalised Markov processes' have applications in computing science for modelling probabilistic programming and speciication languages 7, 4]. We use the resulting calculus of operators over a ...