# Differentiability

• Kai Lai Chung
Chapter
Part of the Die Grundlehren der Mathematischen Wissenschaften book series (volume 104)

## 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 ).

## Keywords

Transition Matrix Continuous Parameter Continuous Derivative Preceding Proof General State Space
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