A Tutorial in Data Science: Lecture 8 – The Functional Theory of Communication in Stochastic Processes

by | Jan 12, 2021 | Math Lecture

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 X_t. A Markov chain is a stochastic process with a fixed historical dependence, in that \mathbb{P}(X_t=i_t |X_{t-1}= i_{t-1}, \cdots , X_0=i_0) =\mathbb{P}(X_t=i_t |X_{t-1}, \cdots , X_{t-c}) for a fixed historical memory length of c. By transforming the single-variable random variable X_t into a c length vector X^{(c)}t=(X_t, X{t-1}, \cdots , X_{t-c+1}) we can translate a historical dependent of c into a historical dependence of 1, by

    \[\mathbb{P}(X^{(c)} t=i^{(c)}_t| X^{(c)}{t-1}=i^{(c)}{t-1}, \cdots ,X^{(c)}{0}=i^{(c)}{0})\]

    \[ =\mathbb{P}\bigg(X_t=i_t, \cdots , X{t-c+1}=i_{t-c+1}| X_{t-1}=i_{t-1}, \cdots , X_{0}=i_{0}\bigg) \]

    \[=\mathbb{P}\bigg(X_t=i_t | X_{t-1}=i_{t-1}, \cdots , X_{0}=i_{0}\bigg) \]

    \[=\mathbb{P}(X_t=i_t |X_{t-1}=i_{t-1}, \cdots , X_{t-c}=i_{t-c})\]

    \[=\mathbb{P}\bigg(X_t=i_t, \cdots , X_{t-c+1}=i_{t-c+1}| X_{t-1}=i_{t-1}, \cdots , X_{t-c}=i_{t-c}\bigg)\]

    \[=\mathbb{P}(X^{(c)}t=i^{(c)}_t| X^{(c)}{t-1}=i^{(c)}_{t-1})\]

The classification of states begins with whether a state occurs an infinite or finite number of times, called recurrent or transient. T_y=\min{n>0: X_n=y} is the time of first return for state-y and T^{(k)}y=\min{n>T^{(k-1)}: X_n=y} is the time of kth return. Let \rho{yy}=\mathbb{P}y(T_y<\infty) =\mathbb{P}(T_y<\infty| X_0=y)=\mathbb{P}(X_{n:0<n<\infty}=y| X_0=y). Clearly, \mathbb{P}y(T^{(k)}_y<\infty) =\mathbb{P}_y(T^{(k)}_y<\infty, T^{(k-1)}_y<\infty) =\mathbb{P}_y(T^{(k)}_y<\infty | T^{(k-1)}_y<\infty)\mathbb{P}_y(T^{(k-1)}_y<\infty) =\mathbb{P}_y(T^{(1)}_y<\infty)\mathbb{P}_y(T^{(k-1)}_y<\infty) by the strong markov property of homogeneity under finite time-translations, thus implying \mathbb{P}_y(T^{(k)}_y<\infty)=\rho{yy}^k. Thus, \underset{k \rightarrow \infty}{\lim}\mathbb{P}y(T^{(k)}_y<\infty) =\underset{k \rightarrow \infty}{\lim}\rho{yy}^k, so the probability of a state recurring an infinite number of times is 0 when \rho_{yy}<1, i.e. it is a transient state, or is 1 when \rho_{yy}=1, i.e. it is a recurrent state.

The communicativity from x to y, i.e. |x \rightarrow y|, is given by \rho_{xy}=\mathbb{P}x(T_y<\infty) =\mathbb{P}(T_y<\infty| X_0=x) =\mathbb{P}(X{n<0} with \rho_{yx}>0.

For intercommunicating x and y cleary, if x is recurrent and communicates with y, then y is also recurrent, since for every ocurrence of x there is a positive probability of transitioning to y. intercommunicates with a recurrent state x, then it is also recurrent. other state then it also recurs, since \rho_{xx}=\mathbb{P}_x(T_x<\infty) =\mathbb{P}(T_x<\infty| X_0=x) =\mathbb{P}(X_{n:0<n<\infty}=x| X_0=x) \leq \mathbb{P}(X_{n:0<n<\infty}=x, X_{m:0<m<n}=y| X_0=x)

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 (t\rightarrow X_t). The measurement action is physically present as an action of cognition whereby an infinitely counted progression of temporality (\lim_{t \to +\infty}X_t) within the imaginary dimension of thought constitutes a complete cycle (1i) of cognitive transcendence between the phenomena and the itself of the state-system: k=\lim_{t \to +\infty} X_t + i. Here, the system is the measuring temporalization (X_t) of the states of being \Omega.

References

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

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