---
title: "Blumenthal's zero–one law"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Blumenthal's_zero–one_law"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T12:21:39.916017+00:00"
instance: "kb-cron"
---

In the mathematical theory of probability, Blumenthal's zero–one law, named after Robert McCallum Blumenthal, is a statement about the nature of the beginnings of right continuous Feller process. Loosely, it states that any right continuous Feller process on 
  
    
      
        [
        0
        ,
        ∞
        )
      
    
    {\displaystyle [0,\infty )}
  
 starting from deterministic point has also deterministic initial movement.


== Statement ==
Suppose that 
  
    
      
        X
        =
        (
        
          X
          
            t
          
        
        :
        t
        ≥
        0
        )
      
    
    {\displaystyle X=(X_{t}:t\geq 0)}
  
 is an adapted right continuous Feller process on a probability space 
  
    
      
        (
        Ω
        ,
        
          
            F
          
        
        ,
        {
        
          
            
              F
            
          
          
            t
          
        
        
          }
          
            t
            ≥
            0
          
        
        ,
        
          P
        
        )
      
    
    {\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t\geq 0},\mathbb {P} )}
  
 such that 
  
    
      
        
          X
          
            0
          
        
      
    
    {\displaystyle X_{0}}
  
 is constant with probability one. Let 
  
    
      
        
          
            
              F
            
          
          
            t
          
          
            X
          
        
        :=
        σ
        (
        
          X
          
            s
          
        
        ;
        s
        ≤
        t
        )
        ,
        
          
            
              F
            
          
          
            
              t
              
                +
              
            
          
          
            X
          
        
        :=
        
          ⋂
          
            s
            >
            t
          
        
        
          
            
              F
            
          
          
            s
          
          
            X
          
        
      
    
    {\displaystyle {\mathcal {F}}_{t}^{X}:=\sigma (X_{s};s\leq t),{\mathcal {F}}_{t^{+}}^{X}:=\bigcap _{s>t}{\mathcal {F}}_{s}^{X}}
  
. Then any event in the germ sigma algebra 
  
    
      
        Λ
        ∈
        
          
            
              F
            
          
          
            0
            +
          
          
            X
          
        
      
    
    {\displaystyle \Lambda \in {\mathcal {F}}_{0+}^{X}}
  
  has either 
  
    
      
        
          P
        
        (
        Λ
        )
        =
        0
      
    
    {\displaystyle \mathbb {P} (\Lambda )=0}
  
 or 
  
    
      
        
          P
        
        (
        Λ
        )
        =
        1.
      
    
    {\displaystyle \mathbb {P} (\Lambda )=1.}
  


== Generalization ==
Suppose that 
  
    
      
        X
        =
        (
        
          X
          
            t
          
        
        :
        t
        ≥
        0
        )
      
    
    {\displaystyle X=(X_{t}:t\geq 0)}
  
 is an adapted  stochastic process on a probability space 
  
    
      
        (
        Ω
        ,
        
          
            F
          
        
        ,
        {
        
          
            
              F
            
          
          
            t
          
        
        
          }
          
            t
            ≥
            0
          
        
        ,
        
          P
        
        )
      
    
    {\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t\geq 0},\mathbb {P} )}
  
 such that 
  
    
      
        
          X
          
            0
          
        
      
    
    {\displaystyle X_{0}}
  
 is constant with probability one. If 
  
    
      
        X
      
    
    {\displaystyle X}
  
 has Markov property with respect to the filtration 
  
    
      
        {
        
          
            
              F
            
          
          
            
              t
              
                +
              
            
          
        
        
          }
          
            t
            ≥
            0
          
        
      
    
    {\displaystyle \{{\mathcal {F}}_{t^{+}}\}_{t\geq 0}}
  
 then any event 
  
    
      
        Λ
        ∈
        
          
            
              F
            
          
          
            0
            +
          
          
            X
          
        
      
    
    {\displaystyle \Lambda \in {\mathcal {F}}_{0+}^{X}}
  
  has either 
  
    
      
        
          P
        
        (
        Λ
        )
        =
        0
      
    
    {\displaystyle \mathbb {P} (\Lambda )=0}
  
 or 
  
    
      
        
          P
        
        (
        Λ
        )
        =
        1.
      
    
    {\displaystyle \mathbb {P} (\Lambda )=1.}
  
 Note that every right continuous Feller process on a probability space 
  
    
      
        (
        Ω
        ,
        
          
            F
          
        
        ,
        {
        
          
            
              F
            
          
          
            t
          
        
        
          }
          
            t
            ≥
            0
          
        
        ,
        
          P
        
        )
      
    
    {\displaystyle (\Omega ,{\mathcal {F}},\{{\mathcal {F}}_{t}\}_{t\geq 0},\mathbb {P} )}
  
  has strong Markov property with respect to the filtration 
  
    
      
        {
        
          
            
              F
            
          
          
            
              t
              
                +
              
            
          
        
        
          }
          
            t
            ≥
            0
          
        
      
    
    {\displaystyle \{{\mathcal {F}}_{t^{+}}\}_{t\geq 0}}
  
.


== References ==