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If the Markov chain is time-homogeneous, then the transition matrix P is the same after each step, so the k-step transition probability can be computed as the k-th power of the transition matrix, P k. If the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution π. [41]
A Markov arrival process is defined by two matrices, D 0 and D 1 where elements of D 0 represent hidden transitions and elements of D 1 observable transitions. The block matrix Q below is a transition rate matrix for a continuous-time Markov chain. [5]
In mathematics, a stochastic matrix is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. [1] [2]: 10 It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix.
For a continuous time Markov chain (CTMC) with transition rate matrix, if can be found such that for every pair of states and π i q i j = π j q j i {\displaystyle \pi _{i}q_{ij}=\pi _{j}q_{ji}} holds, then by summing over j {\displaystyle j} , the global balance equations are satisfied and π {\displaystyle \pi } is the stationary ...
The Markov chain tree theorem considers spanning trees for the states of the Markov chain, defined to be trees, directed toward a designated root, in which all directed edges are valid transitions of the given Markov chain. If a transition from state to state has transition probability ,, then a tree with edge set () is defined to have weight ...
Another discrete-time process that may be derived from a continuous-time Markov chain is a δ-skeleton—the (discrete-time) Markov chain formed by observing X(t) at intervals of δ units of time. The random variables X (0), X (δ), X (2δ), ... give the sequence of states visited by the δ-skeleton.
A family of Markov chains is said to be rapidly mixing if the mixing time is a polynomial function of some size parameter of the Markov chain, and slowly mixing otherwise. This book is about finite Markov chains, their stationary distributions and mixing times, and methods for determining whether Markov chains are rapidly or slowly mixing. [1] [4]
A full metal jacket (FMJ) bullet is a small-arms projectile consisting of a soft core (often lead) encased in an outer shell ("jacket") of harder metal, such as gilding metal, cupronickel, or, less commonly, a steel alloy. A bullet jacket usually allows higher muzzle velocities than bare lead without depositing significant amounts of metal in ...