Intra-individual variation across the lifespan: Results from an Austrian panel study

This article explores intra-individual variation and language change across the lifespan of eight speakers from a small Austrian village. Four phonological variables in two settings (informal conversation vs. formal interview) are traced across longitudinal panel data that span 43 years. The analysis reveals an increase of dialect features (retrograde change), even though apparent-time as well as real-time trend studies indicate dialect loss in the Bavarian speaking parts of Austria. The panel data also indicate that neither the group means at one moment in time nor their averaged changes are representative of the intra-individual variation of any of the eight speakers. Regarding this non-representativity, the article introduces the classical ergodic theorem to variationist sociolinguistics. Evidence will be provided that change across the lifespan of an individual is a non-ergodic process. Thus, it is argued that variationists have to be more cautious when they generalise from group-derived estimates to individual developments and vice versa.

Some comments on sampling of ergodic process, an ergodic theorem and turbulent pressure fluctuations

We present a new advanced mathematics engineering proof for an ergodic theorem for finite time duration signals in the frequency domain (periodograms) which is free from Nyquist interval sampling restriction .We also point out the usefulness of such theorem in the context of a model of random vibrations transmissions (pressure fluctuations )

The law of series

We consider an ergodic process on finitely many states, with positive entropy. Our main result asserts that the distribution function of the normalized waiting time for the first visit to a small (i.e., over a long block) cylinder set B is, for majority of such cylinders and up to epsilon, dominated by the exponential distribution function 1−e−t. That is, the occurrences of B along the time axis can appear either with gap sizes of nearly the exponential distribution (as in an independent and identically distributed process), or they attract each other and create ‘series’. We recall that in [T. Downarowicz, Y. Lacroix and D. Leandri. Spontaneous clustering in theoretical and some empirical stochastic processes. ESAIM Probab. Stat. to appear] it is proved that in a typical (in the sense of category) ergodic process (of any entropy), all cylinders B of selected lengths (such lengths have upper density 1 in ℕ) reveal strong attracting. Combining this with the result of this paper, we obtain, globally in ergodic processes of positive entropy and for long cylinder sets, the prevalence of attracting and deficiency of repelling. This phenomenon resembles what in real life is known as the law of series; the common-sense observation that a rare event, having occurred, has a mysterious tendency to untimely repetitions.

Stability of nonlinear stochastic recursions with application to nonlinear AR-GARCH models

We characterize the Lyapunov exponent and ergodicity of nonlinear stochastic recursion models, including nonlinear AR-GARCH models, in terms of an easily defined, uniformly ergodic process. Properties of this latter process, known as the collapsed process, also determine the existence of moments for the stochastic recursion when it is stationary. As a result, both the stability of a given model and the existence of its moments may be evaluated with relative ease. The method of proof involves piggybacking a Foster-Lyapunov drift condition on certain characteristic behavior of the collapsed process.

Stability of nonlinear stochastic recursions with application to nonlinear AR-GARCH models

We characterize the Lyapunov exponent and ergodicity of nonlinear stochastic recursion models, including nonlinear AR-GARCH models, in terms of an easily defined, uniformly ergodic process. Properties of this latter process, known as the collapsed process, also determine the existence of moments for the stochastic recursion when it is stationary. As a result, both the stability of a given model and the existence of its moments may be evaluated with relative ease. The method of proof involves piggybacking a Foster-Lyapunov drift condition on certain characteristic behavior of the collapsed process.