Summary
Strong approximation theorems for continuous time semimartingales are obtained by combining some techniques of the general theory of stochastic processes with some of the direct approximation of dependent random variables by independent ones. Continuous processes with independent increments whose variance functions increase polynomially or exponentially are considered as approximating processes. The basic assumptions of the main results only contain rates of convergence for certain probabilities. In particular, moment assumptions are not required. Some almost sure invariance principles for partial sum processes with nonlinear growth of variance and for functionals of Markov processes are derived by applying the main results.
Article PDF
Similar content being viewed by others
References
Berkes, I., Philipp, W.: Approximation theorems for independent and weakly dependent random vectors. Ann. Probab.7, 29–54 (1979)
Bhattacharya, R.N.: On the functional central limit theorem and the law of the iterated logarithm for Markov process. Z. Wahrscheinlichkeitstheor. Verw. Geb.60, 185–201 (1982)
Breiman, L.: Probability, Reading, Mass.: Adison-Wesley 1968
Csörgö, M., Révész, P.: Strong approximations in probability and statistics. New York: Academic Press 1981
Dehling, H.: Limit theorems for sums of weakly dependent Banach space valued random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.63, 393–423 (1983)
Dellacherie, C., Meyer, P.A.: Probabilités et potentiel I. Paris: Hermann 1975
Dellacherie, C., Meyer, P.A.: Probabilités et potentiel II. Paris: Hermann 1982
Doob, J.L.: Stochastic processes. New York: Wiley 1953
Dynkin, E.B.: Markov processes. Berlin Heidelberg New York: Springer 1965
Eberlein, E.: On strong invariance principles under dependence assumptions. Ann. Probab.14, 260–270 (1986)
Eberlein, E.: Strong approximation of continuous time stochastic processes. J. Multivariate Anal.31, 220–235 (1989)
Einmahl, U.: Strong invariance principles for partial sums of independent random vectors. Ann. Probab.15, 1419–1440 (1987)
Elliott, R.J.: Stochastic calculus and applications. Berlin Heidelberg New York: Springer 1982
Hall, P., Heyde, C.C.: Martingale limit theory and its applications. New York: Academic Press 1980
Jacod, J.: Calcul stochastique et problèmes des martingales. (Lect. Notes Math., vol. 714) Berlin Heidelberg New York: Springer 1979
Jacod, J.: Théorèmes limite pour les processus. (Lect. Notes Math., vol. 1117, pp. 299–409) Berlin Heidelberg New York: Springer 1983
Jacod, J., Klopotowski, A., Mémin, J.: Théorème de la limite centrale et convergence fonctionelle vers un processus à accroissements indépendants: la méthode des martingales. Ann. Inst. H. Poincaré, Section B,XVIII, 1–45 (1982)
Jacod, J., Shiryaev, A.N.: Limit theorems for stochastic processes. Berlin Heidelberg New York: Springer 1987
Kuelbs, J., Philipp, W.: Almost sure invariance principles for partial sums of mixingB-valued random variables. Ann. Probab.8, 1003–1037 (1980)
Lenglart, E.: Relation de domination entre deux processus. Ann. Inst. H. Poincaré, Section B,XIII, 171–179 (1977)
Liptser, R.Sh., Shiryaev, A.N.: A functional central limit theorem for semimartingales. Theory Probab. Appl.25, 667–688 (1980)
Liptser, R.Sh., Shiryaev, A.N.: On the rate of convergence in the central limit theorem for semimartingales. Theory Probab. Appl.27, 1–13 (1982)
Métivier, M.: Semimartingales: a course on stochastic processes. Berlin, New York: de Gruyter 1982
Morrow, G., Philipp, W.: An almost sure invariance principle for Hilbert space valued martingales. Trans. Am. Math. Soc.273, 231–251 (1982)
Philipp, W.: Invariance principles for independent and weakly dependent random variables. In: Eberlein, E., Taqqu, M. (eds.). Dependence in Probability and Statistics, pp. 225–268. Boston Basel: Birkhäuser 1986
Philipp, W.: A note on the almost sure approximation of weakly dependent random variables. Monatsh. Math.102, 227–236 (1986)
Philipp, W., Stout, W.: Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Am. Math. Soc.161, 1–140 (1975)
Philipp, W., Stout, W.: Invariance principles for martingales and sums of independent random variables. Math. Z.192, 253–264 (1986)
Rebolledo, R.: La méthode des martingales appliquée à l'étude de la convergence en loi de processus. Mem. Soc. Math. Fr.62 (1979)
Shiryaev, A.N.: Martingales: recent developments, results and applications. Int. Stat. Rev.49, 199–233 (1981)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Besdziek, N. Strong approximations of semimartingales by processes with independent increments. Probab. Th. Rel. Fields 87, 489–520 (1991). https://doi.org/10.1007/BF01304277
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01304277