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Convergence of Probability Measures is a graduate textbook in the field of mathematical probability theory. It was written by Patrick Billingsley and published by Wiley in 1968. A second edition in 1999 both simplified its treatment of previous topics and updated the book for more recent developments. [1]
In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than ...
For (,) a measurable space, a sequence μ n is said to converge setwise to a limit μ if = ()for every set .. Typical arrow notations are and .. For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ n of measures on the interval [−1, 1] given by μ n (dx) = (1 + sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.
Patrick Paul Billingsley (May 3, 1925 – April 22, 2011 [1] [2]) was an American mathematician and stage and screen actor, noted for his books in advanced probability theory and statistics. He was born and raised in Sioux Falls, South Dakota , and graduated from the United States Naval Academy in 1946.
Very often, the measures in question are probability measures, so the last part can be written as μ ( K ε ) > 1 − ε . {\displaystyle \mu (K_{\varepsilon })>1-\varepsilon .\,} If a tight collection M {\displaystyle M} consists of a single measure μ {\displaystyle \mu } , then (depending upon the author) μ {\displaystyle \mu } may either ...
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
In probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. It is named after Eduard Helly and Hubert Evelyn Bray. Let F and F 1, F 2, ... be cumulative distribution functions on the real line.
where the last step follows by the pigeonhole principle and the sub-additivity of the probability measure. Each of the probabilities on the right-hand side converge to zero as n → ∞ by definition of the convergence of {X n} and {Y n} in probability to X and Y respectively.