Notes
A unified framework -- Empirical and limiting spectral distribution -- Moment method -- A metric for probability measures -- Patterned matrices: a unified approach -- Scaling -- Reduction to bounded case -- Trace formula and circuits -- Words -- Vertices -- Pair-matched word -- Sub-sequential limit -- Exercises -- Common symmetric patterned matrices -- Wigner matrix -- Semi-circle law, non-crossing partitions, Catalan words -- LSD -- Toeplitz and Hankel matrices -- Toeplitz matrix -- Hankel matrix -- Reverse circulant matrix -- Symmetric circulant and related matrices -- Additional properties of the LSD -- Moments of Toeplitz and Hankel LSD -- Contribution of words and comparison of LSD -- Unbounded support of Toeplitz and Hankel LSD -- Non-unimodality of Hankel LSD -- Density of Toeplitz LSD -- Pyramidal multiplicativity -- Exercises -- Patterned XX matrices -- A unified set up -- Aspect ratio y = -- Preliminaries -- Sample variance-covariance matrix -- Catalan words and Marcenko-Pastur law -- LSD -- Other XX matrices -- Aspect ratio y = -- Sample variance-covariance matrix -- Other XX matrices -- Exercises -- k-circulant matrices -- Normal approximation -- Circulant matrix -- k-circulant matrices -- Eigenvalues -- Eigenvalue partition -- Lower order elements -- Degenerate limit -- Non-degenerate limit -- Exercises -- Wigner type matrices -- Wigner-type matrix -- Exercises -- Balanced Toeplitz and Hankel matrices -- Main results -- Exercises -- Patterned band matrices -- LSD for band matrices -- Proof -- Reduction to uniformly bounded input -- Trace formula, circuits, words and matches -- Negligibility of higher order edges -- (M) condition -- Exercises -- Triangular matrices -- General pattern -- Triangular Wigner matrix -- LSD -- Contribution of Catalan words -- Exercises -- Joint convergence of iid patterned matrices -- Non-commutative probability space -- Joint convergence -- Nature of the limit -- Exercises -- Joint convergence of independent patterned matrices -- Definitions and notation -- Joint convergence -- Freeness -- Sum of independent patterned matrices -- Proofs -- Exercises -- Autocovariance matrix -- Preliminaries -- Main results -- Proofs -- Exercises
