LLN for Quadratic Forms of Long Memory Time Series and Its Applications in Random Matrix Theory

Limiting spectral distribution for large sample covariance matrices with graph-dependent elements

“…As is shown in [30], the above theorem follows from Theorem 2 in [30] and Theorem 7.2.2 with Remark 7.2.6. (4) in [23].…”

“…The most general conditions imposed on x p ensure that the quadratic forms x ⊤ p A p x p weakly concentrate around their expectations up to an error term o(p) with probability 1 − o (1), where A p ∈ C p×p is an arbitrary matrix with the spectral norm A p 1. These conditions were studied in [2], [8], [16], [22], [28], [29], and [30]. As shown in [29], the weak concentration property for specific quadratic forms of x p gives necessary and sufficient conditions for the Marchenko-Pastur theorem [18].…”

Posterior consistency for the spectral density of non-Gaussian stationary time series

“…, X n ) and • s denotes the matrix spectral norm. This is closely related to spectral density estimation (see [48] for some further discussion) and will also be a key element of the below proof of consistency.…”

“…Here the second representation is due to the real and symmetric autocovariance function. The above assumptions also guarantee that a law of large numbers for the quadratic form (X n 1 ) A n X n 1 for all A n s 1 (see Corollary 1 in [48]) where X n 1 now denotes the column vector (X 1 , . .…”

“…Since for any ϕ ∈ Θ, ϕ and ϕ param are both lower bounded away from 0 and upper bounded, the spectral norm of C n (ϕ)Γ −1 per C n (ϕ) is upper bounded uniformly over n ∈ N [21, Theorem 2.1 and Lemma 4.1]. Therefore, by Corollary 1 of Yaskov [48], we have…”