Participant Information
I-Ping Tu from Academia Sinica, Taipei.
Paper: A Transition Probability Tensor and its Application to Block Chain
A sequence of random variables (states) $\{X_1,X_2,\cdots,\}$ can be modeled as a Markov Chain if $P(X_{n+1}|X_1,\cdots,X_n)=P(X_{n+1}|X_n)$ for any integer $n\ge 1$. This can be generalized as a Markov Chain of order $m$. An $m^{th}$ order Markov Chain is defined as $P(X_{n+m}|X_1,\cdots,X_n,\cdots,X_{n+m-1})=P(X_{n+m}|X_n,\cdots,X_{n+m-1})$ holds for any $n\ge 1$. An exact solution for the stationary states is usually solved by constructing a fisrt order Markov Chain accordingly with a huge number of states. The corresponding transition matrix would be a huge sparsity square matrix. A tensor representation has been introduced for the transition probability of high order Markov Chain to delete obvious zero probabilities. With some condition, we could exploit the tensor structure to construct a compact transition matrix to solve the stationary states effeciently. We will show a demonstration on calculating the double spending probability under a dynamic difficulty scheme of a Block Chain.