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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 is linear, except for r = 3, when it is dramatically slow, less than linear (see Bifurcation memory). When the parameter 2 < r < 3, except for the initial values 0 and 1, the fixed point x f 2 = 1 − 1 / r {\displaystyle x_{f2}=1-1/r} is the same as when 1 < r ≤ 2.
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.
Asymptotic expansions often occur when an ordinary series is used in a formal expression that forces the taking of values outside of its domain of convergence. For example, we might start with the ordinary series 1 1 − w = ∑ n = 0 ∞ w n {\displaystyle {\frac {1}{1-w}}=\sum _{n=0}^{\infty }w^{n}}
Lyapunov / Asymptotic / Exponential stability; Rate of convergence ... it can be helpful to organize computations in a chart form, as seen below, to avoid making ...
Two classical techniques for series acceleration are Euler's transformation of series [1] and Kummer's transformation of series. [2] A variety of much more rapidly convergent and special-case tools have been developed in the 20th century, including Richardson extrapolation, introduced by Lewis Fry Richardson in the early 20th century but also known and used by Katahiro Takebe in 1722; the ...
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]
Convergence in distribution is the weakest form of convergence typically discussed, since it is implied by all other types of convergence mentioned in this article. However, convergence in distribution is very frequently used in practice; most often it arises from application of the central limit theorem .