ABSTRACT
Let https://www.w3.org/1998/Math/MathML"> X https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq6383.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be a real-valued random variable on a measurable space https://www.w3.org/1998/Math/MathML"> ( X , σ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq6384.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , and let https://www.w3.org/1998/Math/MathML"> X t , t ∈ N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq6385.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> be an infinite sequence of i.i.d. copies of https://www.w3.org/1998/Math/MathML"> X . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq6386.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Let f be a Borel function and define https://www.w3.org/1998/Math/MathML"> f ‾ n = n - 1 f X 0 + ⋯ + f X n - 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq6387.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> to be the cumulative averages of the sequence https://www.w3.org/1998/Math/MathML"> f X t , t ∈ N . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq6388.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> A fundamental theorem in probability is the law of large numbers: If the first moment of https://www.w3.org/1998/Math/MathML"> f ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq6389.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is finite, then https://www.w3.org/1998/Math/MathML"> f ‾ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq6390.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> converges almost surely to https://www.w3.org/1998/Math/MathML"> E [ f ( X ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq6391.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , i.e., https://www.w3.org/1998/Math/MathML"> P l i m n → ∞ f ‾ n = E [ f ( X ) ] = 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429112638/a69a2ff5-a02c-4831-8fae-bdc86f3c20b4/content/eq6392.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .
