Individual ergodic set

The individual ergodic set is an important set of ergodic theory , a branch of mathematics in the border area between stochastics and the theory of dynamic systems . Alternatively, the individual ergodic set is also called the Birkhoff ergodic set or point-wise ergodic set . It provides a form of the strong law of large numbers for dependent random variables and provides the mathematical basis of the ergodic hypothesis of statistical physics . The theorem was proven in 1931 by George David Birkhoff , after whom it is also named. A compact proof is possible using Hopf's maximal ergodic lemma . In addition, the can -Ergodensatz be derived without much effort from the individual ergodic theorem. ${\ displaystyle {\ mathcal {L}} ^ {p}}$

statement

Let it be an integrable random variable (i.e. it has a finite expectation value ) and a measure-preserving transformation on the underlying probability space (i.e. for all in ). Then the means converge ${\ displaystyle X}$ ${\ displaystyle T}$ ${\ displaystyle (\ Omega, {\ mathcal {A}}, P)}$ ${\ displaystyle P (T ^ {- 1} (A)) = P (A)}$ ${\ displaystyle A}$ ${\ displaystyle {\ mathcal {A}}}$

{\ displaystyle {\ frac {1} {n}} \ sum _ {i = 1} ^ {n} X \ circ T ^ {i-1} (\ omega)}

for almost certain against a random variable . ${\ displaystyle n \ to \ infty}$ ${\ displaystyle Y}$

${\ displaystyle Y}$ can be chosen measurably with respect to the σ-algebra generated by the -invariant sets (i.e. ) and can be represented as a conditional expected value . ${\ displaystyle T}$ ${\ displaystyle A}$ ${\ displaystyle T ^ {- 1} (A) = A}$ ${\ displaystyle {\ mathcal {T}}}$ ${\ displaystyle E [X | {\ mathcal {T}}]}$

If is ergodic , then is almost certainly constant equal to the expected value of . ${\ displaystyle T}$ ${\ displaystyle Y}$ ${\ displaystyle X}$

The example of a stationary process

The random variables ( ) form a stationary stochastic process , i. H. is distributed as . Conversely, any stationary stochastic process can be represented in this way if one assumes that and is of the form . (If this is not the case, can the pictorial space with the size of place of and consider.) Here , and the left shift , the on displayed is the maßerhaltende transformation. ${\ displaystyle Y_ {i} = X \ circ T ^ {i-1}}$ ${\ displaystyle i = 1,2, \ dots}$ ${\ displaystyle (Y_ {2}, Y_ {3}, \ dots)}$ ${\ displaystyle (Y_ {1}, Y_ {2}, \ dots)}$ ${\ displaystyle (Y_ {i}) _ {i \ geq 1}}$ ${\ displaystyle \ Omega = \ mathbb {R} ^ {\ {1,2, \ dots \}}}$ ${\ displaystyle Y_ {i}}$ ${\ displaystyle Y_ {i} (\ omega _ {1}, \ omega _ {2}, \ dots) = \ omega _ {i}}$ ${\ displaystyle \ mathbb {R} ^ {\ {1,2, \ dots \}}}$ ${\ displaystyle (Y_ {1}, Y_ {2}, \ dots)}$ ${\ displaystyle \ Omega}$ ${\ displaystyle P}$ ${\ displaystyle X (\ omega _ {1}, \ omega _ {2}, \ dots) = \ omega _ {1}}$ ${\ displaystyle (\ omega _ {1}, \ omega _ {2}, \ dots)}$ ${\ displaystyle (\ omega _ {2}, \ omega _ {3}, \ dots)}$

If they have a finite expectation, it converges according to the Ergod theorem ${\ displaystyle Y_ {i}}$

{\ displaystyle {\ frac {1} {n}} \ sum _ {i = 1} ^ {n} Y_ {i} (\ omega)}

for almost certain against a random variable . This is everyone's conditional expectation . When ergodicity is present, it is almost certainly constant; H. ${\ displaystyle n \ to \ infty}$ ${\ displaystyle Y}$ ${\ displaystyle E [Y_ {i} | {\ mathcal {T}}]}$ ${\ displaystyle Y_ {i}}$ ${\ displaystyle Y}$