WikiDer > Doob martingale - Vikipediya

Bu maqola mavzu bilan tanish bo'lmaganlar uchun etarli bo'lmagan kontekstni taqdim etadi. Iltimos yordam bering maqolani takomillashtirish tomonidan o'quvchi uchun ko'proq kontekstni taqdim etish. (2011 yil mart) (Ushbu shablon xabarini qanday va qachon olib tashlashni bilib oling)

A Doob martingale (nomi bilan Jozef L. Doob,^[1] a nomi bilan ham tanilgan Levy martingale) a ning matematik konstruktsiyasi stoxastik jarayon bu berilganga yaqinlashadi tasodifiy o'zgaruvchi va ega martingale mulki berilganlarga nisbatan filtrlash. Buni ma'lum vaqtgacha to'plangan ma'lumotlarga asoslanib, tasodifiy o'zgaruvchiga eng yaxshi yaqinlashuvning rivojlanayotgan ketma-ketligi deb hisoblash mumkin.

Summalarni tahlil qilayotganda, tasodifiy yurish, yoki ning boshqa qo'shimcha funktsiyalari mustaqil tasodifiy o'zgaruvchilar, tez-tez ishlatilishi mumkin markaziy chegara teoremasi, katta sonlar qonuni, Chernoffning tengsizligi, Chebyshevning tengsizligi yoki shunga o'xshash vositalar. Farqlari mustaqil bo'lmagan o'xshash ob'ektlarni tahlil qilishda asosiy vositalar martingalalar va Azumaning tengsizligi.^{[tushuntirish kerak]}

Ta'rif

Ruxsat bering ${ displaystyle Y}$ bilan har qanday tasodifiy o'zgaruvchi bo'lishi mumkin ${ displaystyle mathbb {E} [| Y |] < infty}$. Aytaylik ${ displaystyle left {{ mathcal {F}} _ {0}, { mathcal {F}} _ {1}, dots right }}$ a filtrlash, ya'ni ${ displaystyle { mathcal {F}} _ {s} subset { mathcal {F}} _ {t}}$ qachon ${ displaystyle s$. Aniqlang

${ displaystyle Z_ {t} = mathbb {E} [Y mid { mathcal {F}} _ {t}],}$

keyin ${ displaystyle left {Z_ {0}, Z_ {1}, dots right }}$ a martingale,^[2] ya'ni Doob martingale, filtrlashga nisbatan ${ displaystyle left {{ mathcal {F}} _ {0}, { mathcal {F}} _ {1}, dots right }}$.

Buni ko'rish uchun e'tibor bering

• ${ displaystyle mathbb {E} [| Z_ {t} |] = mathbb {E} [| mathbb {E} [Y mid { mathcal {F}} _ {t}] |] leq mathbb {E} [ mathbb {E} [| Y | mid { mathcal {F}} _ {t}]] = mathbb {E} [| Y |] < infty}$;
• ${ displaystyle mathbb {E} [Z_ {t} mid { mathcal {F}} _ {t-1}] = mathbb {E} [ mathbb {E} [Y mid { mathcal {F }} _ {t}] mid { mathcal {F}} _ {t-1}] = mathbb {E} [Y mid { mathcal {F}} _ {t-1}] = Z_ { t-1}}$ kabi ${ displaystyle { mathcal {F}} _ {t-1} subset { mathcal {F}} _ {t}}$.

Xususan, har qanday tasodifiy o'zgaruvchilar ketma-ketligi uchun ${ displaystyle left {X_ {1}, X_ {2}, dots, X_ {n} right }}$ ehtimollik maydoni bo'yicha ${ displaystyle ( Omega, { mathcal {F}}, { text {P}})}$ va funktsiyasi ${ displaystyle f}$ shu kabi ${ displaystyle mathbb {E} [| f (X_ {1}, X_ {2}, dots, X_ {n}) |] < infty}$, birini tanlash mumkin edi

${ displaystyle Y: = f (X_ {1}, X_ {2}, nuqtalar, X_ {n})}$

va filtrlash ${ displaystyle left {{ mathcal {F}} _ {0}, { mathcal {F}} _ {1}, dots right }}$ shu kabi

${ displaystyle { begin {aligned} { mathcal {F}} _ {0} &: = left { phi, Omega right }, { mathcal {F}} _ {t} &: = sigma (X_ {1}, X_ {2}, nuqtalar, X_ {t}), forall t geq 1, end {aligned}}}$

ya'ni ${ displaystyle sigma}$tomonidan yaratilgan algebra ${ displaystyle X_ {1}, X_ {2}, dots, X_ {t}}$. Keyin, Doob martingale ta'rifiga ko'ra, jarayon ${ displaystyle left {Z_ {0}, Z_ {1}, dots right }}$ qayerda

${ displaystyle { begin {aligned} Z_ {0} &: = mathbb {E} [f (X_ {1}, X_ {2}, dots, X_ {n}) mid { mathcal {F} } _ {0}] = mathbb {E} [f (X_ {1}, X_ {2}, dots, X_ {n})], Z_ {t} &: = mathbb {E} [ f (X_ {1}, X_ {2}, nuqtalar, X_ {n}) mid { mathcal {F}} _ {t}] = mathbb {E} [f (X_ {1}, X_ {) 2}, dots, X_ {n}) mid X_ {1}, X_ {2}, dots, X_ {t}], forall t geq 1 end {hizalangan}}}$

Doob martingale hosil qiladi. Yozib oling ${ displaystyle Z_ {n} = f (X_ {1}, X_ {2}, nuqtalar, X_ {n})}$. Ushbu martingale isbotlash uchun ishlatilishi mumkin McDiarmidning tengsizligi.

McDiarmidning tengsizligi

Bayonot^[1]

Mustaqil tasodifiy o'zgaruvchilarni ko'rib chiqing ${ displaystyle X_ {1}, X_ {2}, nuqta X_ {n}}$ ehtimollik maydoni bo'yicha ${ displaystyle ( Omega, { mathcal {F}}, { text {P}})}$ qayerda ${ displaystyle X_ {i} in { mathcal {X}} _ {i}}$ Barcha uchun ${ displaystyle i}$ va xaritalash ${ displaystyle f: { mathcal {X}} _ {1} times { mathcal {X}} _ {2} times cdots times { mathcal {X}} _ {n} rightarrow mathbb {R}}$. Doimiy mavjud deb taxmin qiling ${ displaystyle c_ {1}, c_ {2}, dots, c_ {n}}$ hamma uchun shunday ${ displaystyle i}$,

${ displaystyle { underset {x_ {1}, cdots, x_ {i-1}, x_ {i}, x_ {i} ', x_ {i + 1}, cdots, x_ {n}} { sup}} | f (x_ {1}, dots, x_ {i-1}, x_ {i}, x_ {i + 1}, cdots, x_ {n}) - f (x_ {1}, nuqtalar, x_ {i-1}, x_ {i} ', x_ {i + 1}, cdots, x_ {n}) | leq c_ {i}.}$

(Boshqacha qilib aytganda, ning qiymatini o'zgartirish ${ displaystyle i}$koordinata ${ displaystyle x_ {i}}$ qiymatini o'zgartiradi ${ displaystyle f}$ ko'pi bilan ${ displaystyle c_ {i}}$.) Keyin, har qanday kishi uchun ${ displaystyle epsilon> 0}$,

${ displaystyle { text {P}} (f (X_ {1}, X_ {2}, cdots, X_ {n}) - mathbb {E} [f (X_ {1}, X_ {2}, cdots, X_ {n})] geq epsilon) leq exp left (- { frac {2 epsilon ^ {2}} { sum _ {i = 1} ^ {n} c_ {i } ^ {2}}} o'ng),}$

${ displaystyle { text {P}} (f (X_ {1}, X_ {2}, cdots, X_ {n}) - mathbb {E} [f (X_ {1}, X_ {2}, cdots, X_ {n})] leq - epsilon) leq exp left (- { frac {2 epsilon ^ {2}} { sum _ {i = 1} ^ {n} c_ { i} ^ {2}}} o'ng),}$

va

${ displaystyle { text {P}} (| f (X_ {1}, X_ {2}, cdots, X_ {n}) - mathbb {E} [f (X_ {1}, X_ {2}) , cdots, X_ {n})] | geq epsilon) leq 2 exp left (- { frac {2 epsilon ^ {2}} { sum _ {i = 1} ^ {n} c_ {i} ^ {2}}} o'ng).}$

Isbot

Istalganini tanlang ${ displaystyle x_ {1} ', x_ {2}', cdots, x_ {n} '}$ shundayki, qiymati ${ displaystyle f (x_ {1} ', x_ {2}', cdots, x_ {n} ')}$ har qanday narsa uchun chegaralangan ${ displaystyle x_ {1}, x_ {2}, cdots, x_ {n}}$, tomonidan uchburchak tengsizligi,

${ displaystyle { begin {aligned} & | f (x_ {1}, x_ {2}, cdots, x_ {n}) - f (x_ {1} ', x_ {2}', cdots, x_ {n} ') | leq & | f (x_ {1}, x_ {2}, cdots, x_ {n}) - f (x_ {1}', x_ {2}, cdots, x_ {n}) | & + sum _ {i = 1} ^ {n-1} | f (x_ {1} ', cdots, x_ {i}', x_ {i + 1}, cdots , x_ {n}) - f (x_ {1} ', x_ {2}', cdots, x_ {i} ', x_ {i + 1}', x_ {i + 2}, cdots, x_ { n}) | leq & sum _ {i = 1} ^ {n} c_ {i}, end {hizalanmış}}}$

shunday qilib ${ displaystyle f}$ chegaralangan.

Aniqlang ${ displaystyle Z_ {i}: = mathbb {E} [f (X_ {1}, X_ {2}, cdots, X_ {n}) mid X_ {1}, X_ {2}, cdots, X_ {i}]}$ Barcha uchun ${ displaystyle i geq 1}$ va ${ displaystyle Z_ {0}: = mathbb {E} [f (X_ {1}, X_ {2}, cdots, X_ {n})]}$. Yozib oling ${ displaystyle Z_ {n} = f (X_ {1}, X_ {2}, cdots, X_ {n})}$. Beri ${ displaystyle f}$ Doob martingale ta'rifi bilan chegaralangan, ${ displaystyle left {Z_ {i} right }}$ martingale hosil qiladi. Endi aniqlang${ displaystyle { begin {aligned} U_ {i} & = { underset {x in { mathcal {X}} _ {i}} { sup}} mathbb {E} [f (X_ {1) }, cdots, X_ {n}) mid X_ {1}, cdots, X_ {i-1}, x] - mathbb {E} [f (X_ {1}, cdots, X_ {n} ) mid X_ {1}, cdots, X_ {i-1}], L_ {i} & = { underset {x in { mathcal {X}} _ {i}} { inf} } mathbb {E} [f (X_ {1}, cdots, X_ {n}) mid X_ {1}, cdots, X_ {i-1}, x] - mathbb {E} [f ( X_ {1}, cdots, X_ {n}) mid X_ {1}, cdots, X_ {i-1}]. End {hizalangan}}}$

Yozib oling ${ displaystyle L_ {i} leq Z_ {i} -Z_ {i-1} leq U_ {i}}$ va ${ displaystyle U_ {i}, L_ {i}}$ ikkalasi ham ${ displaystyle { mathcal {F}} _ {i-1}}$-o'lchovli. Bunga qo'chimcha,

${ displaystyle { begin {aligned} U_ {i} -L_ {i} & = { underset {x_ {u} in { mathcal {X}} _ {i}, x_ {l} in { mathcal {X}} _ {i}} { sup}} mathbb {E} [f (X_ {1}, cdots, X_ {n}) mid X_ {1}, cdots, X_ {i- 1}, x_ {u}] - mathbb {E} [f (X_ {1}, cdots, X_ {n}) mid X_ {1}, cdots, X_ {i-1}, x_ {l }] & = { pastki qator {x_ {u} in { mathcal {X}} _ {i}, x_ {l} in { mathcal {X}} _ {i}} { sup} } int _ {{ mathcal {X}} _ {i + 1} times cdots times { mathcal {X}} _ {n}} f (X_ {1}, cdots, X_ {i- 1}, x_ {u}, x_ {i + 1}, cdots, x_ {n}) { text {d}} { text {P}} _ {X_ {i + 1}, cdots, X_ {n} mid X_ {1}, cdots, X_ {t-1}, x_ {u}} (x_ {i + 1}, cdots, x_ {n}) & quad - int _ {{ mathcal {X}} _ {i + 1} times cdots times { mathcal {X}} _ {n}} f (X_ {1}, cdots, X_ {i-1}, x_ {l}, x_ {i + 1}, cdots, x_ {n}) { text {d}} { text {P}} _ {X_ {i + 1}, cdots, X_ {n} o'rtalarida X_ {1}, cdots, X_ {t-1}, x_ {l}} (x_ {i + 1}, cdots, x_ {n}) & = { underset {x_ {u} { mathcal {X}} _ {i}, x_ {l} in { mathcal {X}} _ {i}} { sup}} int _ {{ mathcal {X}} _ {i +1} times cdots times { mathcal {X}} _ {n}} f (X_ {1}, cdots, X_ {i-1}, x_ {u}, x_ {i + 1}, cdots, x_ {n}) { text {d}} { text {P }} _ {X_ {i + 1}, cdots, X_ {n}} (x_ {i + 1}, cdots, x_ {n}) & quad - int _ {{ mathcal {X }} _ {i + 1} times cdots times { mathcal {X}} _ {n}} f (X_ {1}, cdots, X_ {i-1}, x_ {l}, x_ { i + 1}, cdots, x_ {n}) { text {d}} { text {P}} _ {X_ {i + 1}, cdots, X_ {n}} (x_ {i + 1) }, cdots, x_ {n}) & = { underset {x_ {u} in { mathcal {X}} _ {i}, x_ {l} in { mathcal {X}} _ {i}} { sup}} int _ {{ mathcal {X}} _ {i + 1} times cdots times { mathcal {X}} _ {n}} f (X_ {1} , cdots, X_ {i-1}, x_ {u}, x_ {i + 1}, cdots, x_ {n}) & quad -f (X_ {1}, cdots, X_ {i -1}, x_ {l}, x_ {i + 1}, cdots, x_ {n}) { text {d}} { text {P}} _ {X_ {i + 1}, cdots , X_ {n}} (x_ {i + 1}, cdots, x_ {n}) & leq { underset {x_ {u} in { mathcal {X}} _ {i}, x_ {l} in { mathcal {X}} _ {i}} { sup}} int _ {{ mathcal {X}} _ {i + 1} times cdots times { mathcal {X }} _ {n}} c_ {i} { text {d}} { text {P}} _ {X_ {i + 1}, cdots, X_ {n}} (x_ {i + 1} , cdots, x_ {n}) & leq c_ {i} end {aligned}}}$

bu erda mustaqillik tufayli uchinchi tenglik mavjud ${ displaystyle X_ {1}, X_ {2}, cdots, X_ {n}}$. So'ngra Azumaning tengsizligining umumiy shakli ga ${ displaystyle left {Z_ {i} right }}$, bizda ... bor

${ displaystyle { text {P}} (f (X_ {1}, cdots, X_ {n}) - mathbb {E} [f (X_ {1}, cdots, X_ {n})]] geq epsilon) = { text {P}} (Z_ {n} -Z_ {0} geq epsilon) leq exp left (- { frac {2 epsilon ^ {2}} { sum _ {i = 1} ^ {n} c_ {i} ^ {2}}} o'ng).}$

Boshqa tomonga bir tomonlama bog'lanish Azumaning tengsizligini qo'llash orqali olinadi ${ displaystyle left {- Z_ {i} right }}$ va ikki tomonlama chegara quyidagidan kelib chiqadi birlashma bilan bog'liq. ${ displaystyle square}$

Shuningdek qarang

• Konsentratsiyadagi tengsizlik - McDiarmid's va shu kabi bir nechta tengsizliklar haqida qisqacha ma'lumot.

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