A note on the largest eigenvalue of a large dimensional sample covariance matrix

Universality results for the largest eigenvalues of some sample covariance matrix ensembles

“…The first study of the asymptotic behavior of the largest eigenvalue goes back to S. Geman [10]. It was later refined in [2] and [26]. In particular, it is well known that lim N →∞ λ 1 = u c + a.s. if the entries of the random matrix X admit moments up to order 4.…”

“…It is in particular proved therein that the spectral measure µ N = 1 N N i=1 δ λ i a.s. converges as N goes to infinity. Set u c ± = σ 2 (1 ± √ γ) 2 . Then one has that lim N →∞ µ N = ρ M P a.s., where dρ M P (x) dx = (u c + − x)(x − u c − ) 2πxσ 2 1 [u c…”

“…Set u c ± = σ 2 (1 ± √ γ) 2 . Then one has that lim N →∞ µ N = ρ M P a.s., where dρ M P (x) dx = (u c + − x)(x − u c − ) 2πxσ 2 1 [u c…”

“…We assume that the entries X ij , 1 ≤ i ≤ N, 1 ≤ j ≤ p, of the sequence of random matrices X = X N are non-necessarily Gaussian random variables satisfying the following conditions. First, in the complex case, (i){ℜeX i,j , ℑmX i,j : 1 ≤ i ≤ N, 1 ≤ j ≤ p} are real independent random variables, (ii) all these real variables have symmetric laws (thus, E[X 2k+1 2 2 , (iv) all their other moments are assumed to be sub-Gaussian i.e. there exists a constant τ > 0 such that uniformly in i, j and k,…”

Dynamic linear panel regression models with interactive fixed effects

“…In the rest of the paper, we only consider asymptotics in which N and T grow at the same rate; that is, we could equivalently write Geman (1980), Silverstein (1989), Bai, Silverstein, and Yin (1988), Krishnaiah (1988), andLatala (2005). Loosely speaking, we expect the result e = O p ( max(N, T )) to hold as long as the errors e it have mean zero, uniformly bounded fourth moment, and weak time-serial and cross-sectional correlation (in some well-defined sense, see the examples).…”