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In probability theory, a stochastic process has the Markov property if the conditional probability distribution of future states of the process, given the present state, depends only upon the current state, i.e. it is conditionally independent of the past states (the path of the process) given the present state. A process with the Markov property is usually called a Markov process, and may be described as Markovian. Probability theory is the mathematical study of probability. ...
In the mathematics of probability, a stochastic process can be thought of as a random function. ...
In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...
In probability theory, two events A and B are conditionally independent given a third event C precisely if the occurrence or non-occurrence of A and B are independent events in their conditional probability distribution given C. In other words, Two random variables X and Y are conditionally independent given...
In probability theory, a Markov process is a stochastic process characterized as follows: The state at time is one of a finite number in the range . ...
Mathematically, if X(t), t > 0, is a stochastic process, the Markov property states that
Markov processes are typically termed (time-) homogeneous if and otherwise are termed (time-) inhomogeneous (or (time-) nonhomogeneous). Homogeneous Markov processes, usually being simpler than inhomogeneous ones, form the most important class of Markov processes.
In some cases, apparently non-Markovian processes may still have Markovian representations, constructed by expanding the concept of the 'current' and 'future' states. Let X be a non-Markovian process. Then we define a process Y, such that each state of Y represents a time-interval of states of X, i.e. mathematically
If Y has the Markov property, then it is a Markovian representation of X. In this case, X is also called a second-order Markov process. Higher-order Markov processes are defined analogously.
An example of a non-Markovian process with a Markovian representation is a moving average time series. The term moving average is used in different contexts. ...
In statistics and signal processing, a time series is a sequence of data points, measured typically at successive times, spaced apart at uniform time intervals. ...
The most famous Markov processes are Markov chains, but many other processes, including Brownian motion, are Markovian. In mathematics, a (discrete-time) Markov chain, named after Andrei Markov, is a discrete-time stochastic process with the Markov property. ...
An example of 1000 simulated steps of Brownian motion in two dimensions. ...
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