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text Rosenblatt, Joseph Liu, Chaoyuan 2015-09-28T15:19:53Z 2015-09-28T15:19:53Z 10000-01-01 2005 2005 We study certain operators defined by infinite series that describe the nature of convergence of stochastic processes; these include square functions, oscillation operators, and variation operators. The goal is to prove that these operators map Linfinity to BMO and are of strong type (p, p) where 1 < p < infinity for the case that the stochastic processes are Lebesgue differentiation or ergodic averages. In Chapter 2, we prove the appropriate sublinear operator interpolation between the weak type (1, 1) estimate and the strong estimate from Linfinity to BMO. In Chapter 3, we prove that these operators map Linfinity to BMO and are of strong type (p, p) which 1 < p < infinity for Lebesgue derivatives. In Chapter 4, we prove that these operators for ergodic averages are of strong type (p, p) for 1 < p < infinity. In the last chapter, we characterize the strong estimate from Linfinity to L infinity for these operators. We also construct explicit counterexamples to show that the role of BMO is vital since these operators do not map Linfinity to L infinity in general. Made available in DSpace on 2015-09-28T15:19:53Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3202132.pdf: 2866165 bytes, checksum: 16873c131f4f7c5aeede4a2341ad6f92 (MD5) Previous issue date: 2005 Embargo set by: Seth Robbins for item 88138 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 119 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2005. http://hdl.handle.net/2142/86857 (MiAaPQ)AAI3202132 eng Mathematics The Convergence of Lebesgue Derivatives and Ergodic Averages Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.