Suppl. desert set

the suppl. desert set is an important set of the stochastics. It supplies a form of the law of the large numbers for dependent variates and is the mathematical basis of the ergodic theorem.

formulation of the suppl. desert set of Birkhoff

<math> X< /math> is an integrable random variable (i.e. it possesses a finite expectancy value) and <math> T< /math> a measure-receiving transformation on the probability area lying to reason <math> (\ omega, \ mathcal A, P)< /math> (D. h. <math> P (T^ {- 1} (A)) = P (A)< /math> for all <math> A </math> in <math> \ mathcal A< /math> ). Then the means converge

<math> {1 \ over n} \ sum_ {i=1} ^ {n} X \ circ T^ {i-1} (\ omega) </math>

for <math> n \ tons \ infty< /math> nearly surely against a variate Y.

Y can do thereby measurably concerning of that T-invariant quantities of A (i.e. <math> T^ {-) sigma algebra< produced> 1} /math (A) = A <math> \ mathcal T< /math> and leave themselves as conditioned expectancy value math <E> [X are selected|\ mathcal T]< /math> represent.

If T is ergodisch, then is <math> Y< /math> nearly surely constantly equal the expectancy value of X.

the example of a stationary process

the variates <math> Y_i = X \ circ T^ {i-1}< /math> (<math> i = 1, 2,…< /math>) form a stationary stochastic process, i.e. <math> (Y_2, Y_3,…)</math> is as distributed as <math> (Y_1, Y_2,…)</math>. Each stationary stochastic process leaves itself math <(>Y_i) _ {i turned around \ to ge1}< /math> represent in this way, if one assumes that <math> \ omega = \ R^ {\ {1.2,… \}}< /math> and <math> Y_i< /math> of the form <math> Y_i (\ omega_1, \ omega_2,…) = \ omega_i< /math> is. (If this the case is not, can one the image space <math> \ R^ {\ {1.2,… \}}< /math> with the bildmass of <math> (Y_1, Y_2,…)</math> in place of <math> \ omega< /math> and <math> P< /math> regard.) thereby are <math> X (\ omega_1, \ omega_2,…) = \ omega_1< /math>, and the “shifting illustration” <math> \ pi< /math>, under <math> (\ omega_1, \ omega_2,…)</math> on <math> (\ omega_2, \ omega_3,…)</math> , is the measure-receiving transformation is illustrated.

If <math> the Y_i< /math> a finite expectancy value, converged after the suppl. desert set thus math

<{>1 \ over n} \ sum_ {i=1 have} ^ {n} Y_i (\ omega) </math>

for <math> n \ tons \ infty< /math> nearly surely against a variate <math> Y< /math>. <math> Y< /math> the conditioned expectancy value is <math> E [Y_i|Y]< /math> everyone <math> Y_i< /math>. If Ergodizität is present, is <math> Y< /math> nearly surely constantly, i.e.

<math> {1 \ over n} (Y_1 +… + Y_n) \, \ tons \, E [Y_i]< /math> nearly surely (<math> i \ ge1< /math> arbitrary).

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