Estimation of Stability in S. N. Bernshtein’s Theorem

Stability of Bernstein’s Characterization of Gaussian Vectors and a Soft Doubling Argument

“…Theorem 3 considers the local stability for a finite interval around t = 0. Theorem 4 extends Bernstein's Theorem to include local stability, and extends the scalar theory in [33] to vectors. We emphasize that both theorems have a common covariance matrix Q which is not the case in [33] but is important to generalize Theorem 1 and to develop further results for product channels and for broadcast channels.…”

“…Theorem 4 extends Bernstein's Theorem to include local stability, and extends the scalar theory in [33] to vectors. We emphasize that both theorems have a common covariance matrix Q which is not the case in [33] but is important to generalize Theorem 1 and to develop further results for product channels and for broadcast channels. Theorem 5 gives two stability results: one for differential entropy and one for correlation matrices.…”

“…The discussion in [30] describes several other stability metrics, including the Lévy metric [31]. We instead follow [32], [33] (see also [34]) and consider a metric in the characteristic function (c.f.) domain.…”

“…The paper [33] develops the following stability theorem. Let P ϵ be the class of (X 1 , X 2 ) for which X 1 and X 2 are independent and X 1 + X 2 and X 1 − X 2 are ϵ-dependent in the c.f.…”