• Bernoulli
  • Volume 5, Number 4 (1999), 615-639.

On the convergence of Dirichlet processes

François Coquet and Leszek Słomiński

Full-text: Open access


For a given weakly convergent sequence {Xn} of Dirichlet processes we show weak convergence of the sequence of the corresponding quadratic variation processes as well as stochastic integrals driven by the Xn values provided that the condition UTD (a counterpart to the condition UT for Dirichlet processes) holds true. Moreover, we show that under UTD the limit process of {Xn} is a Dirichlet process, too.

Article information

Bernoulli, Volume 5, Number 4 (1999), 615-639.

First available in Project Euclid: 19 February 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Dirichlet process stochastic integral weak convergence


Coquet, François; Słomiński, Leszek. On the convergence of Dirichlet processes. Bernoulli 5 (1999), no. 4, 615--639.

Export citation


  • [1] Aldous, D.J. (1978) Stopping times and tightness. Ann. Probab., 6, 335-340.
  • [2] Bertoin, J. (1986) Les processus de Dirichlet en tant qu'espace de Banach. Stochastics, 18, 155-168.
  • [3] Bertoin, J. (1989) Sur une intégrale pour les processus à á variation bornée. Ann. Probab., 17, 1521-1535.
  • [4] Föllmer, H. (1980) Calcul d'Itô sans probabilités. Séminaire de Probabilités, pp. 144-150. Lecture Notes Math., 850. Berlin: Springer-Verlag.
  • [5] Föllmer, H. (1981) Dirichlet processes. Stochastic Integrals, pp. 476-478. Lecture Notes Math., 851. Berlin: Springer-Verlag.
  • [6] Föllmer, H., Protter, Ph. and Shiryaev, A.N. (1995) Quadratic covariation and an extension of Itô's formula. Bernoulli, 1, 149-169.
  • [7] Jacod, J. (1980) Convergence en loi de semimartingales et variation quadratique Séminaire de Probabilités, pp. 547-560. Lecture Notes Math., 850. Berlin: Springer-Verlag.
  • [8] Jacod, J. and Shiryayev, A.N., (1987) Limit Theorems for Stochastic Processes. Berlin: Springer-Verlag.
  • [9] Jakubowski, A., Mémin, J. and Pagès, G. (1989) Convergence en loi des suites d'intégrales stochastiques sur l'espace D1 de Skorokhod. Probab. Theory Related Fields, 81, 111-137.
  • [10] Kurtz, T.G. and Protter, P. (1991a) Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab., 19, 1035-1070.
  • [11] Kurtz, T.G. and Protter, P. (1991b) Wong-Zakai corrections, random evolutions and simulation schemes for SDE's. Proceedings of the Conference in Honor of Moshe Zakai's 65th Birthday, Stochastic Analysis, Haifa, 1991, pp. 331-346. Boston: Academic Press.
  • [12] Mémin, J. and Slominski, L. (1991) Condition UT et stabilité en loi des solutions d'équations différentielles stochastiques. Séminaire de Probabilités XXV, pp. 162-177. Lecture Notes Math. 1485. Berlin: Springer-Verlag.
  • [13] Rozkosz, A. and Slominski, L. (1998) Extended convergence and Dirichlet processes. Stochastics Stochastics Rep., 65, 79-109.
  • [14] Slominski, L. (1989) Stability of strong solutions of stochastic differential equations. Stochastic Processes Applic., 31, 173-202.
  • [15] Slominski, L. (1996) Stability of stochastic differential equations driven by general semimartingales, Dissert. Math., 349, 1-113.
  • [16] Stricker, C. (1985) Loi de semimartingales et critères de compacité. Séminaire de Probabilités XIX, pp. 209-218. Lecture Notes Math. 1123. Berlin: Springer-Verlag.
  • [17] Stroock, D.W. and Varadhan, S.R.S. (1979) Multidimensional Diffussion Processes. New York: Springer-Verlag.
  • [18] Young, L.C. (1936) An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67, 251-282.
  • [19] Wang, A.T. (1977) Quadratic variation of functionals of Brownian motion. Ann. Probab., 5, 756-769.