gale
Let Ξ½ be a probability measure on Cantor space π, and let sβ[0,β).
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1.
A Ξ½-s-supergale is a function d:{0,1}*β[0,β) that satisfies the condition
d(w)Ξ½(w)sβ₯d(w0)Ξ½(w0)s+d(w1)Ξ½(w1)s (1) for all wβ{0,1}*, the set of all finite strings of 0βs and 1βs (including e, the empty string).
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2.
A Ξ½-s-gale is a Ξ½-s-supergale that satisfies the condition with equality for all wβ{0,1}*.
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3.
A Ξ½-supermartingale
is a Ξ½-1-supergale.
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4.
A Ξ½-martingale is a Ξ½-1-gale.
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5.
An s-supergale is a ΞΌ-s-supergale, where ΞΌ is the uniform probability measure.
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6.
An s-gale is a ΞΌ-s-gale.
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7.
A supermartingale is a 1-supergale.
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8.
A martingale is a 1-gale.
Put in another way, a martingale is a function d:{0,1}*β[0,β) such that, for all wβ{0,1}*, d(w)=(d(w0)+d(w1))/2.
Let d be a Ξ½-s-supergale, where Ξ½ is a probability measure on π and sβ[0,β). We say that d succeeds on a sequence Sβπ if
lim sup |
The success set of is .
succeeds on a language if succeeds on the characteristic sequence of . We say that succeeds strongly 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
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Title | gale |
Canonical name | Gale |
Date of creation | 2013-03-22 16:43:37 |
Last modified on | 2013-03-22 16:43:37 |
Owner | skubeedooo (5401) |
Last modified by | skubeedooo (5401) |
Numerical id | 5 |
Author | skubeedooo (5401) |
Entry type | Definition |
Classification | msc 60G46 |
Classification | msc 60G44 |
Classification | msc 60G42 |
Defines | supergale |
Defines | gale |
Defines | supermartingale |
Defines | succeed |
Defines | succeed strongly |
Defines | success set |
Defines | strong success set |