Search results
Results from the WOW.Com Content Network
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.
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 ...
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 ...
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.
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 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
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.