There has been considerable development of distributed adaptive algorithms in recent years with diffusion algorithms offering very attractive features. But their performance analysis has been severely limited in two ways. Firstly, since the algorithms operate in real time, stability analysis must be done realization-wise but there are no such results. Secondly, almost all analyses to date assume white regressors whereas in practice this is rare. In this work we remedy these limitations using stochastic averaging analysis methods. The key to the analysis is the recognition, for the first time, that the associated error systems have a mixed time-scale structure. Simulations illustrate the new results.

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A large number of important engineering applications require solving stochastic differential equations (SDEs) so that geometrical constraints are satisfied, e.g. the solution has to lie in a matrix Lie group. But the few papers that properly derive numerical schemes for SDEs evolving in Lie groups are in the mathematics literature and not accessible to engineers. The engineering literature is also small but plagued with problems. With this in mind we give a direct accessible derivation of numerical schemes for solving such equations. We do not rely on differential geometry or advanced random process theory. We give simulations in a simple case to show how geometric structure is preserved.