Summary
Subspaces D α, α > 0, of D[0, 1] are defined and given complete metrics d α which are stronger than the Prokhorov metric. The spaces (D α d α) are shown to be separable, and their pre-compact subsets are characterized. A condition which is known to guarantee weak pre-compactness of sets of probability measures over D[0, 1] is shown to also guarantee weak pre-compactness of probability measures over D α for appropriate values of α. Applications are made to the weak convergence of measures induced by stochastic processes, and some examples are included.
Article PDF
Similar content being viewed by others
References
Chentsov, N. N.: Weak convergence of stochastic processes whose trajectories have no discontinuities of the second kind and the “heuristic approach to the KolmogorovSmirnov tests∝. Theor. Probab. Appl. 1, 140–144 (1956).
Kolmogorov, A. N., and B. V. Gnedenko: Limit theorems for sums of independent random variables. Cambridge: Addison-Wesley 1954.
Lamperti, J.: On the convergence of stochastic processes. Trans. Amer. math. Soc. 104, 430–435 (1962).
Loève, M.: Probability theory. Princeton, N.Y.: Van Nostrand 1962.
Parthasarathy, K. R.: Probability measures on metric spaces. New York: Academic Press 1967.
Prokhorov, YU. V.: Convergence of random processes and limit theorems in probability. Theor. Probab. Appl. 1, 157–214 (1956).
Woodroofe, M.: On the maximum deviation of the sample density. Ann. math. Statistics 38, 475–481 (1967).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Woodroofe, M. On the weak convergence of stochastic processes without discontinuities of the second kind. Z. Wahrscheinlichkeitstheorie verw Gebiete 11, 18–25 (1968). https://doi.org/10.1007/BF00538382
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00538382