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Non-asymptotic rates of convergence do not have the common, standard definitions that asymptotic rates of convergence have. Among formal techniques, Lyapunov theory is one of the most powerful and widely applied frameworks for characterizing and analyzing non-asymptotic convergence behavior.
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.
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function f ( n ) as n becomes very large.
An even stronger uniform convergence result for the empirical distribution function is available in the form of an extended type of law of the iterated logarithm. [ 3 ] (p 268 ) See asymptotic properties of the empirical distribution function for this and related results.
The asymptotic distribution can be further characterized in several different ways. First, the central limit theorem states that pointwise, ^ has asymptotically normal distribution with the standard rate of convergence: [2]
3.1 Growth rate. 3.2 Divisibility. 3.3 ... When this equivalence is used to check the convergence of a sum by replacing it with ... the asymptotic expansion of the ...
Lyapunov / Asymptotic / Exponential stability; Rate of convergence ... it can be helpful to organize computations in a chart form, as seen below, to avoid making ...
For example, let f(x) = 6x 4 − 2x 3 + 5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms: 6x 4, −2x 3, and 5. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x ...