Document Type
Lecture
Publication Date
10-18-2019
Abstract
In this talk, we investigate the conditions under which a local limit result holds for the linear random field ��ⱼ = ∑ᵢ∈ℤᵈ ��ᵢεⱼ₋ᵢ, defined on ℤᵈ with absolutely continuous innovations εᵢ that are independent and identically distributed with mean zero and finite variance. Let Γ⁽ᵈ⁾ₙ be a sequence of subsets of ℤᵈ. Define the sum ��ₙ = ∑ⱼ∈Γ⁽ᵈ⁾ₙ ��ⱼ, and let ��²ₙ = Var(��ₙ). Building on Shore’s previous work, we are able to show that the local limit theorem holds for both short and long memory linear random fields in the sense that the sequence of measures √(2π��ₙ)·P(��ₙ ∈ (a, b)) of the interval (a, b) converges to Lebesgue measure. That is, limₙ→∞ √(2π��ₙ)·P(��ₙ ∈ (a, b)) = b − a, assuming minimal and reasonable requirements of the innovations and of the sets Γ⁽ᵈ⁾ₙ. This is joint work with Magda Peligrad and Hailin Sang.
Relational Format
presentation
Recommended Citation
Fortune, Timothy, "Local Limit Theorem for Linear Random Fields" (2019). Probability & Statistics Seminar. 30.
https://egrove.olemiss.edu/math_statistics/30
