On Multivariate Quasi-infinitely Divisible Distributions

On weak convergence of quasi-infinitely divisible laws

“…So the sequence G n (+∞) ∈ ‫,ޒ‬ n ∈ ‫,ގ‬ has a finite limit g(0) that must be real. According to Proposition 6, condition (7) holds with some B ⩽ g(0) + 2M. Using Theorem 5, we get all its assertions.…”

“…First, observe that the function g is measurable, because it is an almost everywhere limit of continuous (hence measurable) functions g n , n ∈ ‫.ގ‬ So we have (16) ‫ޒ‬ g n (t)ρ(t) dt → ‫ޒ‬ g(t)ρ(t) dt, n → ∞ for any function ρ ∈ L 1 ‫. )ޒ(‬ Indeed, due to (7), there exists a constant B 0 > 0 such that |g n (t)| ⩽ ∥G n ∥ ⩽ B 0 for all n ∈ ‫,ގ‬ and convergence (16) holds by the Lebesgue dominated convergence theorem.…”

“…Theorem 5. Let (G n ) n∈‫ގ‬ satisfy (7). Suppose that g n (t) → g(t), n → ∞, for almost all t ∈ ‫ޒ‬ with some function g : ‫ޒ‬ → ‫.ރ‬ Then there exists a function G ∈ V such that ∥G∥ ⩽ B and the equality…”