Supplement to the paper “ Change - Point Estimation in High - Dimensional Markov Random Field Models ”

Let {X(t), 1 ≤ t ≤ T} be a sequence of Rp-valued independent random variables. Let Θ ⊆ Rd be an open, non-empty convex parameter space equipped with the Euclidean inner product 〈·, ·〉, and norm‖ · ‖2. We will also use the `1-norm ‖θ‖1 def = ∑d j=1 |θj |, and the `∞-norm ‖θ‖∞ def = max1≤j≤d |θj |. We assume that there exists a change point τ? ∈ {1, . . . , T − 1}, parameters θ ? , θ (2) ? ∈ Θ, such that for t = 1, . . . , τ?, X (t) ∼ g θ (1) ? , and for t = τ? + 1, . . . , T , X (t) ∼ g θ (2) ? , where g (t) θ (1) ? and g (t) θ (2) ? are probability densities on Rp. The goal is to estimate τ?, θ (1) ? , θ (2) ? . This setting includes the Markov random field setting (our main motivation), where g (t) θ (1) ? and