15 A key element of the above proof is that Q(;Xn)converges uniformly in probability to Qo(0).This is often difficult to prove. A useful condition is given by the following lemma: Lemma8.3:IfX1,.,Xn are i.i.d.p(ax;0o)∈{p(c;0):0∈Θ}，Θ is compact,logp(x;0)is continuous in0 for al0∈Θand all x∈X, and if there exists a function d(x)such that logp(x;0)<d(x)for all0∈Θandx∈X,and Eoc[d(X)】<o,then i.Qo(0)=E0ollogp(X;0)]is continuous in 0 ii.supoeQ(:Xn)-Qo(0)0 Example:Suicide seasonality and von Mises'distribution (in class)15 A key element of the above proof is that Q(θ; Xn) converges uniformly in probability to Q0(θ). This is often difficult to prove. A useful condition is given by the following lemma: Lemma 8.3: If X1,...,Xn are i.i.d. p(x; θ0) ∈ {p(x; θ) : θ ∈ Θ}, Θ is compact, log p(x; θ) is continuous in θ for all θ ∈ Θ and all x ∈ X , and if there exists a function d(x) such that | log p(x; θ)| ≤ d(x) for all θ ∈ Θ and x ∈ X , and Eθ0 [d(X)] < ∞, then i. Q0(θ) = Eθ0 [log p(X; θ)] is continuous in θ ii. supθ∈Θ |Q(θ; Xn) − Q0(θ)| P→ 0 Example: Suicide seasonality and von Mises’ distribution (in class)