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The concept of convergence in probability is used very often in statistics. For example, an estimator is called consistent if it converges in probability to the quantity being estimated. Convergence in probability is also the type of convergence established by the weak law of large numbers.
In probability theory, Kolmogorov's Three-Series Theorem, named after Andrey Kolmogorov, gives a criterion for the almost sure convergence of an infinite series of random variables in terms of the convergence of three different series involving properties of their probability distributions.
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.
For a Bernoulli random variable, the expected value is the theoretical probability of success, and the average of n such variables (assuming they are independent and identically distributed (i.i.d.)) is precisely the relative frequency. This image illustrates the convergence of relative frequencies to their theoretical probabilities.
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]
This theorem follows from the fact that if X n converges in distribution to X and Y n converges in probability to a constant c, then the joint vector (X n, Y n) converges in distribution to (X, c) . Next we apply the continuous mapping theorem , recognizing the functions g ( x , y ) = x + y , g ( x , y ) = xy , and g ( x , y ) = x y −1 are ...
Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric.It was introduced by Charles Stein, who first published it in 1972, [1] to obtain a bound between the distribution of a sum of -dependent sequence of random variables and a standard normal distribution in the Kolmogorov (uniform ...
Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence.