On the ergodic theorems (I) (Generalized ergodic theorems)

Ergodic Theorems

Every one of the important strong limit theorems that we have seen thus far – the strong law of large numbers, the martingale convergence theorem, and the ergodic theorem – has relied in a crucial way on a maximal inequality. This is no accident: it can in fact be shown that a maximal inequality is a necessary condition for an almost everywhere convergence theorem. We will refrain from carrying...

The ergodic theorems

Ergodic Theorems on Amenable Groups

Ergodic Theorems on Amenable Groups

In 1931, Birkhoff gave a general and rigorous description of the ergodic hypothesis from statistical meachanics. This concept can be generalized by group actions of a large class of amenable groups on σ-finite measure spaces. The expansion of this theory culminated in Lindenstrauss’ celebrated proof of the general pointwise ergodic theorem in 2001. The talk is devoted to the introduction of abs...

Ergodic Theorems in Demography

The ergodic theorems of demography describe the properties of a product of certain nonnegative matrices, in the limit as the number of matrix factors in the product becomes large. This paper reviews these theorems and, where possible, their empirical usefulness. The strong ergodic theorem of demography assumes fixed age-specific birth and death rates. An approach to a stable age structure and t...