- Research Summary.
- My main
research interests are probability, statistics and functional analysis. These interests reside in
many areas of mathematics ranging from applied mathematics/statistics to pure mathematics
(probability distributions on abstract spaces). I am extremely interested in research problems that
have applications and require expertise in overlapping areas of mathematics. I have had many
opportunities to work and continue to work on problems from bootstrap, which, of course, overlaps
statistics and limit theorems.
Recently, I have been working on a number of different problems
which I categorize under the following headings: bootstrap, complete convergence, and weak law of
large numbers in Banach spaces of martingale type.
- Bootstrap.
- Bootstrap - a computer-based
method of statistical inference that answers statistical questions without formulas - gives a direct
appreciation of variance, bias, coverage, and other probabilistic phenomena.
For a sequence of
random variables {Xn,n³1}, the exact convergence
rate (i.e., an iterated logarithm type result) is obtained for bootstrapped means. No assumptions are
made concerning either the marginal or joint distributions of the random variables
{Xn,n³1}. As special cases, new results follow for
pairwise i.i.d. sequences and stationary ergodic sequences.
- Complete convergence.
- The concept of complete
convergence was introduced by Hsu and Robbins [4] as follows. A sequence of random variables
{Un,n³1}is said to converge completely to a
constant C if.ån=1¥P{|Un-C|>u}< ¥ for all u>0. In view of the Borel-Cantelli
lemma, this implies that Un®C almost surely (a.s.). The
converse is true if the {Un,n³1} are independent.
Hsu and Robbins [4] proved that the sequence of arithmetic means of independent and identically
distributed (i.i.d.) random variables converges completely to the expected value if the variance of
the summands is finite.
We extend and generalize some recent results on complete convergence (cf. Hu, Moricz, and Taylor
[5], Gut [1] and [2], Kuczmaszewska
and Szynal [7], Pruitt [8], Sung [9], Wang, Bhaskara Rao, and Yang [10]) for arrays of rowwise
independent Banach space valued random
elements. In the main result, no assumptions are made concerning the existence of expected values or
absolute moments of the random elements and no assumptions are made concerning the geometry of the
underlying Banach space. Some well-known results from the literature are obtained easily as
corollaries. The corresponding convergence rates are also established .
- Weak law of large numbers in Banach spaces of martingale
type.
- For weighted sums of the form Sn=åknj-1anj(Vnj-cnj
) where { anj,1£j£kn<¥,n³1} are constants, { Vnj,1£j£kn,n³1} are random elements in a real separable martingale type p Banach
space, and {cnj,1£j£kn,n³1} are suitable
conditional expectations, a mean convergence theorem and a general weak law of large numbers are
established. These results take the form ||Sn||®Lr0 and Sn®P 0, respectively. No conditions are imposed on the joint
distributions of the { Vnj,1£j£kn,n³1}. The mean
convergence theorem is proved assuming that {|| Vnj||r,1£j£kn,n³1} is {|anj|r}-uniformly
integrable whereas the weak law is proved
under a Ces\`{a}ro type condition which is weaker than Ces\`{a}ro uniform integrability. The sharpness
of the results is illustrated by an example. These results extend that of Gut [3] and Hong and Oh [6].
