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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 ...
Hawkes process; Homogeneous processes: processes where the domain has some symmetry and the finite-dimensional probability distributions also have that symmetry. Special cases include stationary processes, also called time-homogeneous. Karhunen–Loève theorem; Lévy process; Local time (mathematics) Loop-erased random walk
A Markov chain with two states, A and E. In probability, a discrete-time Markov chain (DTMC) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past.
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.
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.
Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity.