Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T14:58:41.186Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak convergence of sequences of random elements with random indices

Published online by Cambridge University Press:  24 October 2008

M. Csörgő  and
Z. Rychlik
M. Csörgő
Affiliation:
Carleton University and MariaCurie-Sklodowska University, Lublin
Z. Rychlik
Affiliation:
Carleton University and MariaCurie-Sklodowska University, Lublin
Get access Rights & Permissions [Opens in a new window]

Extract

Let (S, d) be a separable metric space equipped with its Borel σ field . Let {Yn, n ≥ 1} be a sequence of S-valued random elements defined on a probability space (Ω, , p). Assume Yn ⇒ Y converges weakly to an S-valued random element Y. Let {Nn, n ≥ 1} be a sequence of positive integer-valued random variables defined on the same probability space (Ω, , p).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 1 , July 1980 , pp. 171 - 174
Copyright
Copyright © Cambridge Philosophical Society 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Aldous, D. J.Weak convergence of randomly indexed sequences of random variables. Math. Proc. Cambridge Philos. Soc. 83 (1978), 117–126.CrossRefGoogle Scholar
(2)Aldous, D. J. and Eagleson, G. K.On mixing and stability of limit theorems. Ann. Probability 6 (1978), 325–331.CrossRefGoogle Scholar
(3)Anscombe, F. J.Large-sample theory of sequential estimation. Proc. Cambridge Philos. Soc. 48 (1952), 600–607.CrossRefGoogle Scholar
(4)Babu, G. J. and Ghosh, M.A random functional central limit theorems for martingales. Acta Math. Acad. Sci. Hung. 27 (1976), 301–306.CrossRefGoogle Scholar
(5)Durrett, R. T. and Resnick, S. I.Weak convergence with random indices. Stochastic Processes and their Appl. 5 (1977), 213–220.CrossRefGoogle Scholar
(6)Guiasu, S.On the asymptotic distribution of sequences of random variables with random indices. Ann. Math. Statist. 42 (1971), 2018–2028.CrossRefGoogle Scholar
(7)Hall, P.On the Skorokhod representation approach to martingale invariance principles. Ann. Probability 7 (1979), 371–376.CrossRefGoogle Scholar
(8)Lindvaal, T.Weak convergence of probability measures and random functions in the function space D[0, ∞). J. Appl. Prob. 10 (1973), 109–121.CrossRefGoogle Scholar
(9) Prakasa Rao, B. L. S.Limit theorems for random number of random elements on complete separable metric spaces. Acta Math. Acad. Sci. Hungar. 24 (1973), 1–4.CrossRefGoogle Scholar
(10)Renyi, A.On mixing sequences of sets. Acta Math. Acad. Sci. Hungar. 9 (1958), 215–228.CrossRefGoogle Scholar
(11)Rychlik, Z.The order of approximation in the random central limit theorem. Lect. Notes in Math. no. 656 (1978), 225–236.CrossRefGoogle Scholar