Seminar on Stochastic Processes, 1988 pp 99-107 | Cite as
Reminiscences of some of Paul Lévy’s ideas in Brownian Motion and in Markov Chains
Chapter
Abstract
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 . It is known that p’ ij (0) = q ij exists and 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.
$$\mathop {\lim }\limits_{t \downarrow 0} p_{ii} (t) = 1 $$
(1)
$$ 0 \leqslant {q_i} = - {q_{{ii}}} \leqslant + \infty, \;0 \leqslant {q_{{ij}}} < \infty, i \ne j; $$
(2)
$$\sum\limits_{{j \ne i}} {{q_{{ij}}} \leqslant {q_i}.} $$
(3)
Keywords
Markov Chain Brownian Motion Local Time Sojourn Time Occupation Time
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Preview
Unable to display preview. Download preview PDF.
Copyright information
© Birkhäuser Boston 1989