In an ergodic system, the crossstream (note the artificially induced sss) and downstream properties are the same. Coin tossing games are not that way. The properties of 10 games of matching pennies with 10 tosses are not the same as that of a single game of 100 tosses. (Trivially, one cannot be behind 11 points in a 10 toss game.)
Aye. And for non-ergodic, they are not.
The issue I was trying to raise was the possibility whether
a) there are "almost" ergodic systems, where depending on the property observed, you can treat them as though they really were ergodic, without too big an error...
b) are there cases where one thinks something is ergodic, and it turns out it *wasn't* at all? How far up the proverbial creek does one end up?
Just stirring the pot...
Cheers!