What does donsker's theorem mean?
Definitions for donsker's theorem
donsker's the·o·rem
This dictionary definitions page includes all the possible meanings, example usage and translations of the word donsker's theorem.
Wikipedia
Donsker's theorem
In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem. Let X 1 , X 2 , X 3 , … {\displaystyle X_{1},X_{2},X_{3},\ldots } be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Let S n := ∑ i = 1 n X i {\displaystyle S_{n}:=\sum _{i=1}^{n}X_{i}} . The stochastic process S := ( S n ) n ∈ N {\displaystyle S:=(S_{n})_{n\in \mathbb {N} }} is known as a random walk. Define the diffusively rescaled random walk (partial-sum process) by W ( n ) ( t ) := S ⌊ n t ⌋ n , t ∈ [ 0 , 1 ] . {\displaystyle W^{(n)}(t):={\frac {S_{\lfloor nt\rfloor }}{\sqrt {n}}},\qquad t\in [0,1].} The central limit theorem asserts that W ( n ) ( 1 ) {\displaystyle W^{(n)}(1)} converges in distribution to a standard Gaussian random variable W ( 1 ) {\displaystyle W(1)} as n → ∞ {\displaystyle n\to \infty } . Donsker's invariance principle extends this convergence to the whole function W ( n ) := ( W ( n ) ( t ) ) t ∈ [ 0 , 1 ] {\displaystyle W^{(n)}:=(W^{(n)}(t))_{t\in [0,1]}} . More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space D [ 0 , 1 ] {\displaystyle {\mathcal {D}}[0,1]} , the random function W ( n ) {\displaystyle W^{(n)}} converges in distribution to a standard Brownian motion W := ( W ( t ) ) t ∈ [ 0 , 1 ] {\displaystyle W:=(W(t))_{t\in [0,1]}} as n → ∞ . {\displaystyle n\to \infty .}
Numerology
Chaldean Numerology
The numerical value of donsker's theorem in Chaldean Numerology is: 9
Pythagorean Numerology
The numerical value of donsker's theorem in Pythagorean Numerology is: 9
Translation
Find a translation for the donsker's theorem definition in other languages:
Select another language:
- - Select -
- 简体中文 (Chinese - Simplified)
- 繁體中文 (Chinese - Traditional)
- Español (Spanish)
- Esperanto (Esperanto)
- 日本語 (Japanese)
- Português (Portuguese)
- Deutsch (German)
- العربية (Arabic)
- Français (French)
- Русский (Russian)
- ಕನ್ನಡ (Kannada)
- 한국어 (Korean)
- עברית (Hebrew)
- Gaeilge (Irish)
- Українська (Ukrainian)
- اردو (Urdu)
- Magyar (Hungarian)
- मानक हिन्दी (Hindi)
- Indonesia (Indonesian)
- Italiano (Italian)
- தமிழ் (Tamil)
- Türkçe (Turkish)
- తెలుగు (Telugu)
- ภาษาไทย (Thai)
- Tiếng Việt (Vietnamese)
- Čeština (Czech)
- Polski (Polish)
- Bahasa Indonesia (Indonesian)
- Românește (Romanian)
- Nederlands (Dutch)
- Ελληνικά (Greek)
- Latinum (Latin)
- Svenska (Swedish)
- Dansk (Danish)
- Suomi (Finnish)
- فارسی (Persian)
- ייִדיש (Yiddish)
- հայերեն (Armenian)
- Norsk (Norwegian)
- English (English)
Word of the Day
Would you like us to send you a FREE new word definition delivered to your inbox daily?
Citation
Use the citation below to add this definition to your bibliography:
Style:MLAChicagoAPA
"donsker's theorem." Definitions.net. STANDS4 LLC, 2024. Web. 1 May 2024. <https://www.definitions.net/definition/donsker%27s+theorem>.
Discuss these donsker's theorem definitions with the community:
Report Comment
We're doing our best to make sure our content is useful, accurate and safe.
If by any chance you spot an inappropriate comment while navigating through our website please use this form to let us know, and we'll take care of it shortly.
Attachment
You need to be logged in to favorite.
Log In