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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]
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.
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 ...
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.
The converse, however, is false; i.e., local convergence in measure is strictly weaker than global convergence in measure, in general. If, however, () < or, more generally, if f and all the f n vanish outside some set of finite measure, then the distinction between local and global convergence in measure disappears.
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures.It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces.
The convergence of the sequence and domination by can be relaxed to hold only -almost everywhere i.e. except possibly on a measurable set of -measure . In fact we can modify the functions f n {\displaystyle f_{n}} (hence its point wise limit f {\displaystyle f} ) to be 0 on Z {\displaystyle Z} without changing the value of the integrals.
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.