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A second-order Markov chain can be introduced by considering the current state and also the previous state, as indicated in the second table. Higher, n th-order chains tend to "group" particular notes together, while 'breaking off' into other patterns and sequences occasionally.
Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] A stationary Gauss–Markov process is unique [citation needed] up to rescaling; such a process is also known as an Ornstein–Uhlenbeck process.
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to ...
In the mathematical theory of stochastic processes, variable-order Markov (VOM) models are an important class of models that extend the well known Markov chain models. In contrast to the Markov chain models, where each random variable in a sequence with a Markov property depends on a fixed number of random variables, in VOM models this number of conditioning random variables may vary based on ...
If a stochastic process is N-th-order stationary, then it is also M-th-order stationary for all . If a stochastic process is second order stationary (=) and has finite second moments, then it is also wide-sense stationary. [1]: p. 159
Second order fluid queues (sometimes called Markov modulated diffusion processes or fluid queues with Brownian noise [42]) consider a reflected Brownian motion with parameters controlled by a Markov process. [22] [43] Two different types of boundary conditions are commonly considered: absorbing and reflecting. [44]
The "Markov" in "Markov decision process" refers to the underlying structure of state transitions that still follow the Markov property. The process is called a "decision process" because it involves making decisions that influence these state transitions, extending the concept of a Markov chain into the realm of decision-making under uncertainty.
A process with this property is said to be Markov or Markovian and known as a Markov process. Two famous classes of Markov process are the Markov chain and Brownian motion. Note that there is a subtle, often overlooked and very important point that is often missed in the plain English statement of the definition: the statespace of the process ...