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This Concept Map, created with IHMC CmapTools, has information related to: Chapter 4 - P Richardson, stochastic processes shown as <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mmultiscripts> <mtext> X </mtext> <mtext> n </mtext> <none/> </mmultiscripts> <mtext> , n = 0,1,2,3 . . . </mtext> </mrow> </math>, limiting probabilties shown as matricies, MARKOV CHAINS can be time reversable, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mmultiscripts> <mtext> X </mtext> <mtext> n </mtext> <none/> </mmultiscripts> <mtext> = i </mtext> </math> then the process is said to be "in state i at time n", MARKOV CHAINS are stochastic processes, a one-step transition probability can become an n step transition probability, matricies which have intermediate states known as <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> transient states S = I + PtS </mtext> </mrow> </math>, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mmultiscripts> <mtext> X </mtext> <mtext> n </mtext> <none/> </mmultiscripts> <mtext> , n = 0,1,2,3 . . . </mtext> </mrow> </math> if <math xmlns="http://www.w3.org/1998/Math/MathML"> <mmultiscripts> <mtext> X </mtext> <mtext> n </mtext> <none/> </mmultiscripts> <mtext> = i </mtext> </math>, stochastic processes and are denoted by the set of nonnegative integers {0,1,2,3. . . }, the process is said to be "in state i at time n" or a one-step transition probability, an n step transition probability which can be computed by <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> Chapman-Kolmogorov Equations </mtext> <mmultiscripts> <sum/> <mtext> k=0 </mtext> <mtext> ∞ </mtext> </mmultiscripts> <mtext> </mtext> <mmultiscripts> <mtext> P </mtext> <mtext> ik </mtext> <mtext> n </mtext> </mmultiscripts> <mtext> For all n, m>= 0, all i,j </mtext> </mrow> </math>, MARKOV CHAINS develop into limiting probabilties