Abstract:
The scaled standard Wigner matrix and its limiting eigenvalue distribution,
namely the semi-circular distribution, has attracted much attention. The 2kth
moment of the limit equals the number of non-crossing pair-partitions of the set
f1; 2; : : : ; 2kg. There are several extensions of this result in the literature.
Patterned random matrices such as the reverse circulant, the symmetric circulant,
the Toeplitz and the Hankel matrices and their almost sure limiting spectral
distribution (LSD), have also been studied quite extensively. Under the assumption
that the entries are taken from an i.i.d. sequence with finite variance, the
LSD are tied together by a common thread—the 2kth moment of the limit equals
a weighted sum over di erent types of pair-partitions of the set f1; 2; : : : ; 2kg and
are universal. Some results are also known for the sparse case.
We discuss extension of these results by relaxing significantly the i.i.d. assumption.
With suitable assumption on the entries, the limits are defined via a
larger class of partitions and are also not universal. In the process we show how
some new sets of partitions gain importance in finding the moments of the limits.
Several existing and new results forWigner and other patterned matrices, their
band and sparse versions, as well as for matrices with continuous and discrete
variance profile follow as special cases.
We also look into similar generalisation of existing results for XXT matrices
with independent entries that satisfy certain moment conditions. In this case too,
some new results are generated besides obtaining the existing results as special
cases.
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