page_title: Donsker's theorem
hierarchical_section_title: Donsker's theorem
caption_attribution_description: English: Animation of Donsker's invariance principle.
context_page_description: In probability theory, Donsker's theorem, named after Monroe D. Donsker, is a functional extension of the central limit theorem.
  Let be a sequence of independent and identically distributed random variables with mean 0 and variance 1. Let. The stochastic process is known as a random walk. Define the diffusively rescaled random walk by
  The central limit theorem asserts that converges in distribution to a standard Gaussian random variable as. Donsker's invariance principle extends this convergence to the whole function. More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space, the random function converges in distribution to a standard Brownian motion as
context_section_description: 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  be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Let . The stochastic process  is known as a random walk. Define the diffusively rescaled random walk (partial-sum process) by
  The central limit theorem asserts that  converges in distribution to a standard Gaussian random variable  as . Donsker's invariance principle extends this convergence to the whole function . More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space , the random function  converges in distribution to a standard Brownian motion  as
caption: Donsker's invariance principle for simple random walk on 
    
      
        
          
            Z
          
        
      
      {\displaystyle \mathbb {Z} }
    
  .