Mathematical Sciences: Limit Theorems for Function-Indexed  Empirical Processes
{'Code': '03040107', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'William Y. Velez', 'PO_EMAI': None, 'PO_PHON': None}
The first problem this research will consider is that of                   limit theorems for empirical processes.  Considerable progress                  has been made in recent years towards solving the problem of                    necessary and sufficient conditions on probability measures and                 classes of functions for which the corresponding empirical                      process satisfies the central limit theorem.  Some of these                     advances were made by the investigator himself.  He will continue               to look for the final solution of the problem.                                       The second problem concerns the law of iterated logarithms                 for Banach space valued random variables.  A set of three                       conditions is found to be both necessary and sufficient but one                 of these conditions amounts to proving a weaker version of the                  original problem.  The investigator will attack this reduced                    problem.                                                                             The third and smaller part of the research will deal with                  percolation theory.  Among the topics to be considered will be                  continuity of the percolation probability, bounds on critical                   exponents and finiteness of the expected size of cluster                        containing zero.
{'FirstName': 'Kenneth', 'LastName': 'Alexander', 'PI_MID_INIT': 'S', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Kenneth S Alexander', 'EmailAddress': 'alexandr@usc.edu', 'NSF_ID': '000432683', 'StartDate': '06/15/1987', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Southern California', 'CityName': 'LOS ANGELES', 'ZipCode': '90033', 'PhoneNumber': '2137407762', 'StreetAddress': '3720 S FLOWER ST FL 3', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'California', 'StateCode': 'CA', 'CONGRESSDISTRICT': '34', 'CONGRESS_DISTRICT_ORG': 'CA34', 'ORG_UEI_NUM': 'G88KLJR3KYT5', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF SOUTHERN CALIFORNIA', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '126300', 'Text': 'PROBABILITY'}
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