Ornstein isomorphism theorem

In probability and statistics a Markov renewal process is a random process that generalizes the notion of Markov jump processes. Other random processes like Markov chain, Poisson process, and renewal process can be derived as a special case of an MRP (Markov renewal process).

Definition

Consider a state space $\mathrm{S}.$ Consider a set of random variables $(X_n,T_n)$, where $T_n$ are the jump times and $X_n$ are the associated states in the Markov chain (see Figure). Let the inter-arrival time, ${\tau }_n=T_n-T_{n-1}$. Then the sequence (X_n, T_n) is called a Markov renewal process if

$\mathrm{Pr}({\tau }_{n+1}\le t,X_{n+1}=j|(X_0,T_0),(X_1,T_1),\dots ,(X_n=i,T_n))$

$=\mathrm{Pr}({\tau }_{n+1}\le t,X_{n+1}=j|X_n=i)\mathrm{\forall }n\ge 1,t\ge 0,i,j\in \mathrm{S}$

An illustration of a Markov renewal process

Relation to other stochastic processes

1. If we define a new stochastic process $Y_t:=X_n$ for $t\in [T_n,T_{n+1})$, then the process $Y_t$ is called a semi-Markov process. Note the main difference between an MRP and a semi-Markov process is that the former is defined as a two-tuple of states and times, whereas the latter is the actual random process that evolves over time. The entire process is not Markovian, i.e., memoryless, as happens in a CTMC. Instead the process is Markovian only at the specified jump instants. This is the rationale behind the name, Semi-Markov.
2. A semi-Markov process where all the holding times are exponentially distributed is called a continuous time Markov chain/process (CTMC). In other words, if the inter-arrival times are exponentially distributed and if the waiting time in a state and the next state reached are independent, we have a CTMC.

   $\mathrm{Pr}({\tau }_{n+1}\le t,X_{n+1}=j|(X_0,T_0),(X_1,T_1),\dots ,(X_n=i,T_n))=\mathrm{Pr}({\tau }_{n+1}\le t,X_{n+1}=j|X_n=i)$

   $=\mathrm{Pr}(X_{n+1}=j|X_n=i)(1-e^{-{\lambda }_{i}t}),\text{ for all }n\ge 1,t\ge 0,i,j\in \mathrm{S}$

3. The sequence $X_n$ in the MRP is a discrete-time Markov chain. In other words, if the time variables are ignored in the MRP equation, we end up with a DTMC.

   $\mathrm{Pr}(X_{n+1}=j|X_0,X_1,\dots ,X_n=i)=\mathrm{Pr}(X_{n+1}=j|X_n=i)\mathrm{\forall }n\ge 1,i,j\in \mathrm{S}$

4. If the sequence of $\tau$s are independent and identically distributed, and if their distribution does not depend on the state $X_n$, then the process is a renewal process. So, if the exact states are ignored and we have a chain of iid times, then we have a renewal process.

   $\mathrm{Pr}({\tau }_{n+1}\le t|T_0,T_1,\dots ,T_n)=\mathrm{Pr}({\tau }_{n+1}\le t)\mathrm{\forall }n\ge 1,\mathrm{\forall }t\ge 0$

See also

• Markov process
• Renewal theory
• Variable-order Markov model

References and Further Reading

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