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On Rates of Convergence in Some Strong Limit Theorems for Banach Space-Valued Random Variables

Alf, Carol J.
http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1566297702070788

Abstract Details

1975, Doctor of Philosophy (Ph.D.), Bowling Green State University, Mathematics.
Let (Omega, Delta, P) be a probability space and let B be a separable Banach Space. Let [Xn;n>1] be a sequence of independent random variables taking values in B. Under certain conditions Beck has shown that the laws of large numbers hold. It is therefore of interest to consider the rates of convergence of series with terms such as P[||Sn||>n l/r e], where Sn=[n1 XK, r > 0, and E > O. Convergence equivalence of several such series is established. The methods used here are natural extensions and modifications of those used in the literature for the real-valued case, as for example, done by Rohatgi. This work naturally leads to the consideration of weighted sums of B-valued random variables, Sn=[k=1 an,Kxk, where [an,k] forms an array of real numbers which satisfy certain conditions. Laws of large numbers for these weighted sums are established, which are generalizations of the real-valued case as done by Jamison, Orey, and Pruitt. The method of proof is similar to that used by Mourier in her proof of the strong law of large numbers for independent, identically distributed random elements. Assured of the convergence of weighted sums, we are again led to the study of rates of convergence of terms such as P[||SN\\>e]. The results obtained are natural extensions of the result in the real-valued case done by Hanson and Wright, and Rohtagi. And finally, large deviation probabilities, as defined by Heyde, are considered for sums of B-valued random variables. These results extend those of Andersen in the real-valued case, with some modifications needed in the proofs.
Vijay K. Rohatgi (Advisor)
Mathematics

Recommended Citations

Citations

  • Alf, C. J. (1975). On Rates of Convergence in Some Strong Limit Theorems for Banach Space-Valued Random Variables [Doctoral dissertation, Bowling Green State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1566297702070788

    APA Style (7th edition)

  • Alf, Carol. On Rates of Convergence in Some Strong Limit Theorems for Banach Space-Valued Random Variables. 1975. Bowling Green State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1566297702070788.

    MLA Style (8th edition)

  • Alf, Carol. "On Rates of Convergence in Some Strong Limit Theorems for Banach Space-Valued Random Variables." Doctoral dissertation, Bowling Green State University, 1975. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1566297702070788

    Chicago Manual of Style (17th edition)

bgsu1566297702070788
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