Dynkin system

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set ${\displaystyle \Omega }$ 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 ${\displaystyle D}$ be a collection of subsets of Ω (i.e., ${\displaystyle D}$ is a subset of the power set of Ω). Then ${\displaystyle D}$ is a Dynkin system if

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

Equivalently, ${\displaystyle D}$ is a Dynkin system if

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

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

Dynkin's π-λ theorem

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

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 ${\displaystyle X\colon (\Omega ,{\mathcal {F}},\operatorname {P} )\rightarrow \mathbb {R} }$ in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable is defined as

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

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

${\displaystyle {\mathcal {L}}_{X}(B)=\operatorname {P} \left[X^{-1}(B)\right]}$ for all ${\displaystyle B\in {\mathcal {B}}(\mathbb {R} ),}$

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

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 ${\displaystyle (\Omega ,{\mathcal {F}},\operatorname {P} ),}$ with respectively generated π-systems ${\displaystyle {\mathcal {I}}_{X}}$ and ${\displaystyle {\mathcal {I}}_{Y}.}$ The joint cumulative distribution function of ${\displaystyle (X,Y)}$ is

${\displaystyle F_{X,Y}(a,b)=\operatorname {P} \left[X\leq a,Y\leq b\right]=\operatorname {P} \left[X^{-1}((-\infty ,a])\cap Y^{-1}((-\infty ,b])\right],}$ for all ${\displaystyle a,b\in \mathbb {R} .}$

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

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

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

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

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

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

See also

• Algebra of sets – Identities and relationships between sets involving complements, inclusions ⊆, and finite unions ∪ and intersections ∩.
• δ-ring
• Field of sets – Algebraic concept in measure theory
• Monotone class
• π-system – A non-empty family of sets where the intersection of any two members is again a member.
• Ring of sets
• σ-algebra
• σ-ring

Notes

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

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.

This article incorporates material from Dynkin system on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.