From Laplace’s Calculus of Probability (Lecture 3), a finite difference equation explains probabilistically how a game evolves over discrete time, defining thus a stochastic process and specifically a Markov Chain where there is finite recursion or limited historical dependence. While one can solve these finite difference equations with Taylor polynomials as generating functions when they are infinitely differentiable, when they are not, a Fourier or Laplacian transformation via trigonometric generating functions must be used. If there was instead a continuous time then a differential equation would describe its evolution, which could also be studied recursively in terms of the approximative iterations of its flows as solution domains.

A stochastic process is a time-evolving random process. It is given by a random variable . A Markov chain is a stochastic process with a fixed historical dependence, in that

The classification of states begins with whether a state occurs an infinite or finite number of times, called **recurrent** or **transient**. *time of first return* for state-*time of kth return*. Let **transient** state, or is 1 when **recurrent** state.

The communicativity from

For intercommunicating

Communicativity is a form of an explanation of probability without assuming any knowledge of deterministic causality. That two states of a system communication implies that the system may transition from at least one to other (uni-directional), or between them (bi-direction), over the forward progression of the measurement action’s internal time (

**References**

Richard Durrett, *Essentials of Stochastic Processes*, 3rd Edition, pp.13-20.

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