DES - Demonstration of the Second Property of the Transition Function Matrix

Arguments

• DES - Definition of ‘Transition Function Matrix for CTHMC’

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Given the Matrix H(t) where $p_{i,j}(t)$ is defined as:

$$p_{i,j}(t)\triangleq P(X(s+t)=j\mid X(s)=i)$$

Then the sum along the $i$-th row of $H(t)$ is:

$$\sum _{\genfrac{}{}{0ex}{}{}{j\in \mathcal{X}}}{p}_{i,j}(t)=\sum _{\genfrac{}{}{0ex}{}{}{j\in \mathcal{X}}}P(X(s+t)=j\mid X(s)=i)$$

Where we can define:

$$\begin{array}{rl}1& =\\ & =P(X(s+t)\in \mathcal{X}\mid X(s)=i)\\ & \text{< knowing that: }\mathcal{X}=\bigcup _{\genfrac{}{}{0ex}{}{}{k}}\{k\}\text{>}\\ & =P\left(\bigcup _{\genfrac{}{}{0ex}{}{}{k}}\left\{X(s+t)=j\right\}\mid X(s)=i\right)\\ & =\sum _{\genfrac{}{}{0ex}{}{}{j\in \mathcal{X}}}P(X(s+t)=j\mid X(s)=i)\end{array}$$

The probability that at time $s+t$ (in the future) the state is one of those defined in $\mathcal{X}$ is $1$. It remains $1$ even if starting from one specific state $X(s)=i$

Now, $X(s+t)\in \mathcal{X}$ is defined as the union of all possible states, in this case denoted $X(s+t)=j$

Also this is always equal to say the sum over all possible state, because this is a probability distribution.