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The process is named after the statistician David Cox, who first published the model in 1955. [ 1 ] Cox processes are used to generate simulations of spike trains (the sequence of action potentials generated by a neuron ), [ 2 ] and also in financial mathematics where they produce a "useful framework for modeling prices of financial instruments ...
The term random function is also used to refer to a stochastic or random process, [25] [26] because a stochastic process can also be interpreted as a random element in a function space. [27] [28] The terms stochastic process and random process are used interchangeably, often with no specific mathematical space for the set that indexes the ...
In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as ). That is, it is a function f ( x ) {\displaystyle f(x)} that takes on a random value at each point x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} (or some other domain).
A Bernoulli process is a finite or infinite sequence of independent random variables X 1, X 2, X 3, ..., such that for each i, the value of X i is either 0 or 1; for all values of , the probability p that X i = 1 is the same. In other words, a Bernoulli process is a sequence of independent identically distributed Bernoulli trials.
Complete spatial randomness (CSR) describes a point process whereby point events occur within a given study area in a completely random fashion. It is synonymous with a homogeneous spatial Poisson process. [1] Such a process is modeled using only one parameter , i.e. the density of points within the defined area. The term complete spatial ...
This random process finds wide application in model building: In physics , spin systems and fluorescence intermittency show dichotomous properties. But especially in single molecule experiments probability distributions featuring algebraic tails are used instead of the exponential distribution implied in all formulas above.
The mathematical definition of ergodicity aims to capture ordinary every-day ideas about randomness.This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as diffusion and Brownian motion, as well as common-sense notions of mixing, such as mixing paints, drinks, cooking ingredients, industrial process mixing, smoke in a smoke-filled room, the ...
Consider a stochastic process X : [0, T] × Ω → R, and equip the real line R with its usual Borel sigma algebra generated by the open sets.. If we take the natural filtration F • X, where F t X is the σ-algebra generated by the pre-images X s −1 (B) for Borel subsets B of R and times 0 ≤ s ≤ t, then X is automatically F • X-adapted.