Let be a probability measure on Cantor space , and let .
A --gale is a --supergale that satisfies the condition with equality for all .
A -supermartingale is a -1-supergale.
A -martingale is a -1-gale.
An -supergale is a --supergale, where is the uniform probability measure.
An -gale is a --gale.
A supermartingale is a 1-supergale.
A martingale is a 1-gale.
Put in another way, a martingale is a function such that, for all , .
Let be a --supergale, where is a probability measure on and . We say that succeeds on a sequence if
The strong success set of is .
Intuitively, a supergale is a betting strategy that bets on the next bit of a sequence when the previous bits are known. is the parameter that tunes the fairness of the betting. The smaller is, the less fair the betting is. If succeeds on a sequence, then the bonus we can get from applying as the betting strategy on the sequence is unbounded. If succeeds strongly on a sequence, then the bonus goes to infinity.
|Date of creation||2013-03-22 16:43:37|
|Last modified on||2013-03-22 16:43:37|
|Last modified by||skubeedooo (5401)|
|Defines||strong success set|