Differentiability

Abstract

Let (p _ij ) be a standard transition matrix. The derivatives at zero of the p _ij established in the preceding section are of basic importance in the study of the associated Markov chain. The following notation so will be used throughout the rest of this monograph:

$${{q}_{i}}=-p{{\prime }_{ii}}\left( 0 \right),~ {{q}_{ij}}=p{{\prime }_{ij}}\left( 0 \right),~i\ne j$$

(1)

foil Occasionally the notation q _ii =−q _i will also be used; the matrix will then be called the Q-matrix of the matrix

$$\left( {{q}_{ij}} \right)=\left( p{{\prime }_{ij}}\left( 0 \right) \right)$$

will then be called the Q-matrix of the matrix(p _ij ).