Canonical Equation K ₃₃ for the Fourier Transform of the Resolvent of a Gram Block Random Matrix

Abstract

In this chapter, certain random Gram matrices H _n with stationary (in wide sense) random entries ξ _ij are considered. This direction of investigation of the limit of the normalized spectral functions (eigenvalue counting functions)

$$ {\mu _n}(x,\;{H_n}) = {n^{ - 1}}\sum\limits_{k = 1}^n \chi \;\{ {\lambda _i}({H_n})\;{\rm{ < }}x\} $$

was developed in [Weg], [BKV], [PaK] for random Gram matrices

$$ {H_n} = \left\{ {\left. {b(i - j) + {n^{ - 1}}\sum\limits_{k = 1}^p {{\xi _{ik}}{\xi _{jk}}} } \right\}_{i,j = 1}^n} \right. $$

with dependent random entries ξ _jk . In these papers, it has been assumed that the random variables ξ _ik have a joint Gaussian distribution with the properties Eξ _ik = 0, Eξ _ik ξ _jp = V _i-j (k - p), where the function V _j (x) is such that V _-j (-x) = V _j (x),

$$ \sum\limits_{j,k = - \infty }^\infty {|{V_j}(k)|} = V < \infty $$

and the nonrandom sequence b(k), (k = 0, ± 1, . . .) satisfies the condition

$$ b\;( - k) = b\;(k),\;\sum\limits_{k = - \infty }^\infty {\;|b\;(k)|} < \infty . $$