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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]
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.
More explicitly, let P n (ε) be the probability that X n is outside the ball of radius ε centered at X. Then X n is said to converge in probability to X if for any ε > 0 and any δ > 0 there exists a number N (which may depend on ε and δ) such that for all n ≥ N, P n (ε) < δ (the definition of limit).
The weak limit of a sequence of probability measures, provided it exists, is a probability measure. In general, if tightness is not assumed, a sequence of probability (or sub-probability) measures may not necessarily converge vaguely to a true probability measure, but rather to a sub-probability measure (a measure such that μ ( X ) ≤ 1 ...
For instance, a risk-neutral measure is a probability measure which assumes that the current value of assets is the expected value of the future payoff taken with respect to that same risk neutral measure (i.e. calculated using the corresponding risk neutral density function), and discounted at the risk-free rate.
The concept of probability function is made more rigorous by defining it as the element of a probability space (,,), where is the set of possible outcomes, is the set of all subsets whose probability can be measured, and is the probability function, or probability measure, that assigns a probability to each of these measurable subsets .
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 measure corresponding to a CDF is said to be induced by the CDF. This measure coincides with the pmf for discrete variables and PDF for continuous variables, making the measure-theoretic approach free of fallacies. The probability of a set in the σ-algebra is defined as