Reminiscences of some of Paul Lévy’s ideas in Brownian Motion and in Markov Chains
Kai Lai Chung · Birkhäuser Boston eBooks · 1989
We begin with a resume. Let {P(t), t ≥ 0} be a semigroup of stochastic matrices with elements p ij (t), (i,j) ∈ I ×I, where I is a countable set, satisfying the condition $$\mathop {\lim }\limits_{t \downarrow 0} p_{ii} (t) = 1 $$ (1). It is known that p’ ij (0) = q ij exists and $$ 0 \leqslant {q_i} = - {q_{{ii}}} \leqslant + \infty, \;0 \leqslant {q_{{ij}}} < \infty, i e j; $$ (2) $$\sum\limits_{{j e i}} {{q_{{ij}}} \leqslant {q_i}.} $$ (3) The state i is called stable if q i < +∞, and instantaneous if q i = +∞ (Lévy’s terminology). The matrix Q = (q ij ) is called conservative when equality holds in (3) for all i.