We introduce kernel estimators for the semi-circular law. In this first part of our general theory on the estimators, we prove the consistency and conduct simulation study to show the performance of the estimators. We also point out that Wigner's semi-circular law for our new estimators and the classical empirical spectral distributions is still true when the elements of Wigner matrices do not have finite variances but are in the domain of attraction of normal law.
Received 25 May 2012Accepted 18 February 2013Published online 11 March 2013
W. Zhou. was in part supported by a grant R-155-000-131-112 at the National University of Singapore. The author also would like to thank one referee for reading the manuscript carefully and pointing out several typos.
Article outline: I. INTRODUCTION II. METHODOLOGY AND MAIN RESULTS III. SIMULATION STUDY IV. PROOF OF THEOREMS