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Recurrent sequences +:= (), called fixed point iterations, define discrete time autonomous dynamical systems and have important general applications in mathematics through various fixed-point theorems about their convergence behavior.
The different possible notions of convergence relate to how such a behavior can be characterized: two readily understood behaviors are that the sequence eventually takes a constant value, and that values in the sequence continue to change but can be described by an unchanging probability distribution.
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 = / is the same as when 1 < r ≤ 2. However, in this case the convergence is not monotonically.
Convergence in total variation is stronger than weak convergence. An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".
The rate of return expected by such an investor is equal to the relative entropy between the investor's believed probabilities and the official odds. [20] This is a special case of a much more general connection between financial returns and divergence measures.
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.
A common formulation of the martingale convergence theorem for discrete-time martingales is the following. Let X 1 , X 2 , X 3 , … {\displaystyle X_{1},X_{2},X_{3},\dots } be a supermartingale. Suppose that the supermartingale is bounded in the sense that
Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to