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Reminiscences of some of Paul Lévy’s ideas in Brownian Motion and in Markov Chains

  • Kai Lai Chung
Chapter
Part of the Progress in Probability book series (PRPR, volume 17)

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
$$\mathop {\lim }\limits_{t \downarrow 0} p_{ii} (t) = 1 $$
(1)
. It is known that p’ ij (0) = q ij exists and
$$ 0 \leqslant {q_i} = - {q_{{ii}}} \leqslant + \infty, \;0 \leqslant {q_{{ij}}} < \infty, i \ne j; $$
(2)
$$\sum\limits_{{j \ne i}} {{q_{{ij}}} \leqslant {q_i}.} $$
(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.

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.

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Copyright information

© Birkhäuser Boston 1989

Authors and Affiliations

  • Kai Lai Chung

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