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单词 Gale
释义

gale


Let ν be a probability measureMathworldPlanetmath on Cantor space 𝐂, and let s[0,).

  1. 1.

    A ν-s-supergale is a function d:{0,1}*[0,) that satisfies the condition

    d(w)ν(w)sd(w0)ν(w0)s+d(w1)ν(w1)s(1)

    for all w{0,1}*, the set of all finite strings of 0’s and 1’s (including e, the empty string).

  2. 2.

    A ν-s-gale is a ν-s-supergale that satisfies the condition with equality for all w{0,1}*.

  3. 3.

    A ν-supermartingalePlanetmathPlanetmath is a ν-1-supergale.

  4. 4.

    A ν-martingale is a ν-1-gale.

  5. 5.

    An s-supergale is a μ-s-supergale, where μ is the uniform probability measure.

  6. 6.

    An s-gale is a μ-s-gale.

  7. 7.

    A supermartingale is a 1-supergale.

  8. 8.

    A martingale is a 1-gale.

Put in another way, a martingale is a function d:{0,1}*[0,) such that, for all w{0,1}*, d(w)=(d(w0)+d(w1))/2.

Let d be a ν-s-supergale, where ν is a probability measure on 𝐂 and s[0,). We say that d succeeds on a sequence S𝐂 if

lim supnd(S[0..n-1])=.

The success set of d is S[d]={S𝐂|d succeeds on S}.d succeeds on a languagePlanetmathPlanetmath A{0,1}* if d succeeds on the characteristic sequence χA of A. We say that d succeeds strongly on a sequence S𝐂 if

lim infnd(S[0..n-1])=.

The strong success set of d is Sstr[d]={S𝐂|d succeeds strongly on S}.

Intuitively, a supergale d is a betting strategy that bets on the next bit of a sequence when the previous bits are known. s is the parameter that tunes the fairness of the betting. The smaller s is, the less fair the betting is. If d succeeds on a sequence, then the bonus we can get from applying d as the betting strategy on the sequence is unboundedPlanetmathPlanetmath. If d succeeds strongly on a sequence, then the bonus goes to infinityMathworldPlanetmath.

Titlegale
Canonical nameGale
Date of creation2013-03-22 16:43:37
Last modified on2013-03-22 16:43:37
Ownerskubeedooo (5401)
Last modified byskubeedooo (5401)
Numerical id5
Authorskubeedooo (5401)
Entry typeDefinition
Classificationmsc 60G46
Classificationmsc 60G44
Classificationmsc 60G42
Definessupergale
Definesgale
Definessupermartingale
Definessucceed
Definessucceed strongly
Definessuccess set
Definesstrong success set
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