On the convergence of Dirichlet processes. (English) Zbl 0953.60001

\(X\) is called a Dirichlet process if there exist processes \(M, A\) such that \(X=M+A\) where \(M\) is a local martingale and \(A\) is an adapted process of 0-quadratic variation along some sequence \((D_k)_k\) of partitions of \([0,T]\) with \(\max_{t_j \in D_k} |t_{j+1}-t_j|\longrightarrow 0\) as \(k \rightarrow \infty\), i.e. \(\sum_{t_j \in D_k} |\triangle A_{t_j}|^2 \longrightarrow 0\) in probability as \(k \rightarrow \infty\). For sequences of continuous Dirichlet processes, it is introduced a condition UTD, which is a counterpart of condition UT for semimartingales. Under UTD condition some stability theorems for continuous Dirichlet processes and for stochastic integrals driven by continuous Dirichlet processes are established. It is proved that under UTD condition the limit process of Dirichlet processes is also a Dirichlet process. Functionals of Dirichlet processes are investigated. J. Bertoin’s result [Ann. Probab. 17, No. 4, 1521-1535 (1989; Zbl 0687.60054)] on the existence of a stochastic integral \(\int_0^t X_sdY_s\) for Dirichlet processes \(X, Y\) is slightly generalized.


60F05 Central limit and other weak theorems


Zbl 0687.60054
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