Summary
In [4], Sen proved Theorems 22.1 and 22.2 of Billingsley [1] on the weak convergence of empirical distribution functions for sequences of Φ-mixing random variables to appropriate Gaussian random functions under less stringent regularity conditions. In this note, we shall prove the same theorems under weaker conditions than Sen's ones.
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References
Billingsley, P.: Convergence of probability measures. New York: Wiley (1968)
Ibragimov, I.A.: Some limit theorems for stationary processes. Theory Probability Appl. 7, 349–382 (1962)
Oodaira, H., Yoshihara, K.: The law of the iterated for stationary processes satisfying mixing conditions. Kodai Math. Sem. Rep. 23, 311–334 (1971)
Sen, P.K.: A note on weak convergence of empirical processes for sequences of Φ-mixing random variables. Ann. Math. Statist. 42, 2131–2133 (1971)
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Yoshihara, K.i. Extensions of Billingsley's theorems on weak convergence of empirical processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 29, 87–92 (1974). https://doi.org/10.1007/BF00533190
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DOI: https://doi.org/10.1007/BF00533190