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This definition is technically called Q-convergence, short for quotient-convergence, and the rates and orders are called rates and orders of Q-convergence when that technical specificity is needed. § R-convergence , below, is an appropriate alternative when this limit does not exist.
The definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, and even to the “random variables” which are not measurable — a situation which occurs for example in the study of empirical processes. This is the “weak convergence of laws without laws being ...
The rate of convergence is distinguished from the number of iterations required to reach a given accuracy. For example, the function f ( x ) = x 20 − 1 has a root at 1. Since f ′(1) ≠ 0 and f is smooth, it is known that any Newton iteration convergent to 1 will converge quadratically.
The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets. [2] The Glivenko–Cantelli classes arise in Vapnik–Chervonenkis theory, with applications to machine learning. Applications can be found in econometrics making use of M-estimators.
This definition is strictly dependent on the norm used in the space; the choice of such norm is fundamental to estimate the rate of convergence and, in general, all ...
The rate of convergence must be chosen carefully, though, usually h ∝ n −1/5. In many cases, highly accurate results for finite samples can be obtained via numerical methods (i.e. computers); even in such cases, though, asymptotic analysis can be useful. This point was made by Small (2010, §1.4), as follows.
The order in probability notation is used in probability theory and statistical theory in direct parallel to the big O notation that is standard in mathematics.Where the big O notation deals with the convergence of sequences or sets of ordinary numbers, the order in probability notation deals with convergence of sets of random variables, where convergence is in the sense of convergence in ...
This is the pointwise (in θ) convergence. A particular example of a uniform law of large numbers states the conditions under which the convergence happens uniformly in θ. If [29] [30] Θ is compact, f(x,θ) is continuous at each θ ∈ Θ for almost all xs, and measurable function of x at each θ.