Abstract
| For measure preserving ergodic dynamical systems there is no strong law of large
numbers if the expectation has infinite mean.
In some cases it is possible to obtain a generalised strong law of large
numbers by deleting the maximum entry, for examples for the digits of a
continued fraction expansion.
Haynes proved that there is no strong law of large numbers by only deleting a
finite number of terms (light trimming) on iterations of the doubling map with
observable $1/x$, eventhough this would be possible if one considered
independent random variables with the same distribution function.
Based on this result Haynes and myself have shown that a strong law does still
hold in this case depending if for the number of deleted terms $b_n$ of the sum
of the first $n$ entries it holds that $\lim_{n\to\infty}b_n/\log\log\log
n=\infty$ or not. |