Dynkin system

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set $\Omega$ satisfying a set of axioms weaker than those of σ-algebra. Dynkin systems are sometimes referred to as λ-systems (Dynkin himself used this term) or d-system.[1] These set families have applications in measure theory and probability.

A major application of λ-systems is the π-λ theorem, see below.

Definitions

Let Ω be a nonempty set, and let $D$ be a collection of subsets of Ω (i.e., $D$ is a subset of the power set of Ω). Then $D$ is a Dynkin system if

Ω ∈ $D$ ,
if A, B ∈ $D$ and A ⊆ B, then B $\setminus$ A ∈ $D$ ,
if A₁, A₂, A₃, ... is a sequence of subsets in $D$ and A_n ⊆ A_n+1 for all n ≥ 1, then $\bigcup _{n=1}^{\infty }A_{n}\in D$ .

Equivalently, $D$ is a Dynkin system if

Ω ∈ $D$ ,
if A ∈ ${\textstyle D}$ , then A^c ∈ $D$ ,
if A₁, A₂, A₃, ... is a sequence of subsets in $D$ such that A_i ∩ A_j = Ø for all i ≠ j, then $\bigcup _{n=1}^{\infty }A_{n}\in D$ .

The second definition is generally preferred as it usually is easier to check.

An important fact is that a Dynkin system which is also a π-system (i.e., closed under finite intersections) is a σ-algebra. This can be verified by noting that conditions 2 and 3 together with closure under finite intersections imply closure under countable unions.

Given any collection ${\mathcal {J}}$ of subsets of $\Omega$ , there exists a unique Dynkin system denoted $D\{{\mathcal {J}}\}$ which is minimal with respect to containing ${\mathcal {J}}$ . That is, if ${\tilde {D}}$ is any Dynkin system containing ${\mathcal {J}}$ , then $D\{{\mathcal {J}}\}\subseteq {\tilde {D}}$ . $D\{{\mathcal {J}}\}$ is called the Dynkin system generated by ${\mathcal {J}}$ . Note $D\{\emptyset \}=\{\emptyset ,\Omega \}$ . For another example, let $\Omega =\{1,2,3,4\}$ and ${\mathcal {J}}=\{1\}$ ; then $D\{{\mathcal {J}}\}=\{\emptyset ,\{1\},\{2,3,4\},\Omega \}$ .

Dynkin's π-λ theorem

If $P$ is a π-system and $D$ is a Dynkin system with $P\subseteq D$ , then $\sigma \{P\}\subseteq D$ . In other words, the σ-algebra generated by $P$ is contained in $D$ .

One application of Dynkin's π-λ theorem is the uniqueness of a measure that evaluates the length of an interval (known as the Lebesgue measure):

Let (Ω, B, λ) be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let μ be another measure on Ω satisfying μ[(a,b)] = b − a, and let D be the family of sets S such that μ[S] = λ[S]. Let I = { (a,b),[a,b),(a,b],[a,b] : 0 < a ≤ b < 1 }, and observe that I is closed under finite intersections, that I ⊂ D, and that B is the σ-algebra generated by I. It may be shown that D satisfies the above conditions for a Dynkin-system. From Dynkin's π-λ Theorem it follows that D in fact includes all of B, which is equivalent to showing that the Lebesgue measure is unique on B.

Application to probability distributions

The $π$ -λ theorem motivates the common definition of the probability distribution of a random variable $X\colon (\Omega ,{\mathcal {F}},\operatorname {P} )\rightarrow \mathbb {R}$ in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable is defined as

F_{X}(a)=\operatorname {P} [X\leq a],\qquad a\in \mathbb {R} ,

whereas the seemingly more general law of the variable is the probability measure

{\mathcal {L}}_{X}(B)=\operatorname {P} \left[X^{-1}(B)\right]

for all

B\in {\mathcal {B}}(\mathbb {R} ),

where ${\mathcal {B}}(\mathbb {R} )$ is the Borel 𝜎-algebra. We say that the random variables $X\colon (\Omega ,{\mathcal {F}},\operatorname {P} ),$ and $Y\colon ({\tilde {\Omega }},{\tilde {\mathcal {F}}},{\tilde {\operatorname {P} }})\rightarrow \mathbb {R}$ (on two possibly different probability spaces) are equal in distribution (or law), denoted by $X\,{\stackrel {\mathcal {D}}{=}}\,Y,$ if they have the same cumulative distribution functions, that is, $F_{X}=F_{Y}.$ The motivation for the definition stems from the observation that if $F_{X}=F_{Y},$ then that is exactly to say that ${\mathcal {L}}_{X}$ and ${\mathcal {L}}_{Y}$ agree on the $π$ -system $\left\{(-\infty ,a]\colon a\in \mathbb {R} \right\}$ which generates ${\mathcal {B}}(\mathbb {R} ),$ and so by the example above: ${\mathcal {L}}_{X}={\mathcal {L}}_{Y}.$

A similar result holds for the joint distribution of a random vector. For example, suppose X and Y are two random variables defined on the same probability space $(\Omega ,{\mathcal {F}},\operatorname {P} ),$ with respectively generated $π$ -systems ${\mathcal {I}}_{X}$ and ${\mathcal {I}}_{Y}.$ The joint cumulative distribution function of $(X,Y)$ is

F_{X,Y}(a,b)=\operatorname {P} \left[X\leq a,Y\leq b\right]=\operatorname {P} \left[X^{-1}((-\infty ,a])\cap Y^{-1}((-\infty ,b])\right],

for all

a,b\in \mathbb {R} .

However, $A=X^{-1}((-\infty ,a])\in {\mathcal {I}}_{X}$ and $B=Y^{-1}((-\infty ,b])\in {\mathcal {I}}_{Y}.$ Because

{\mathcal {I}}_{X,Y}=\left\{A\cap B:A\in {\mathcal {I}}_{X},{\text{ and }}B\in {\mathcal {I}}_{Y}\right\}

is a $π$ -system generated by the random pair $(X,Y),$ the $π$ -λ theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of $(X,Y).$ In other words, $(X,Y)$ and $(W,Z)$ have the same distribution if and only if they have the same joint cumulative distribution function.

In the theory of stochastic processes, two processes $(X_{t})_{t\in T},(Y_{t})_{t\in T}$ are known to be equal in distribution if and only if they agree on all finite-dimensional distributions. i.e. for all $t_{1},\ldots ,t_{n}\in T,\,n\in \mathbb {N} .$

\left(X_{t_{1}},\ldots ,X_{t_{n}}\right)\,{\stackrel {\mathcal {D}}{=}}\,\left(Y_{t_{1}},\ldots ,Y_{t_{n}}\right).

The proof of this is another application of the $π$ -λ theorem.[2]

Notes

Aliprantis, Charalambos; Border, Kim C. (2006). Infinite Dimensional Analysis: a Hitchhiker's Guide (Third ed.). Springer. Retrieved August 23, 2010.
Kallenberg, Foundations Of Modern probability, p. 48

References

Gut, Allan (2005). Probability: A Graduate Course. New York: Springer. doi:10.1007/b138932. ISBN 0-387-22833-0.
Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. ISBN 0-471-00710-2.
Williams, David (2007). Probability with Martingales. Cambridge University Press. p. 193. ISBN 0-521-40605-6.

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[1] Aliprantis, Charalambos; Border, Kim C. (2006). Infinite Dimensional Analysis: a Hitchhiker's Guide (Third ed.). Springer. Retrieved August 23, 2010.

[2] Kallenberg, Foundations Of Modern probability, p. 48