Pairwise independence

Basic Concepts in Probability

Published in X. Rong Li, Probability, Random Signals, and Statistics, 2017

In general, if n events are independent then they are pairwise independent. But the fact that they are pairwise independent does not imply they are independent. This is a surprise to many people. For a set of events, if none of the events individually will affect the occurrence of any other event in the set, then these events are pairwise independent. They are independent if none of the events individually or collectively will affect the occurrence of any other event in the set. For example, P{ABC} = P{A}P{B}P{C} neither implies nor is implied by the following P{AB}=P{A}P{B}P{BC}=P{B}P{C}P{CA} = P{C}P{A}

Estimates for the volume variation of compact submanifolds driven by a stochastic flow

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Journal Information

Published in Dynamical Systems, 2022

Diego Sebastian Ledesma, Robert Andres Galeano Anaya, Fabiano Borges da Silva

More precisely, in this work, we consider a compact n-dimensional submanifold N of a Riemannian m-dimensional manifold M with the metric h, and an SDE where are smooth vector fields on M, and are pairwise independent Brownian motions on a filtered probability space . We assume that the SDE generates a solution flow of diffeomorphisms (such as in Baxendale [2,3] and Kunita [22]), and for each we denote the random submanifold, obtained through the action of the stochastic flow , by . As usual, we omit the ω most of the time and just write .

Pairwise independence

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Basic Concepts in Probability

Estimates for the volume variation of compact submanifolds driven by a stochastic flow