and (B.20) at m = V ( π$_{1}$ ) establishes (i) : We argue (ii) by contraposition. Suppose that V ( π$_{1}$ ) < V ( π$_{2}$ ) − c . By the monotonicity of cumulative distribution functions, F$_{π}$$_{2}$ ( V ( π$_{1}$ )) ≤ F$_{π}$$_{2}$ ( V ( π$_{2}$ ) − c ). Combining this with (B.20) at m = V ( π$_{2}$ ) − c , By (B.19) and the fact that F$_{π}$$_{2}$ ( V ( π$_{2}$ )) = F$_{π}$$_{1}$ ( V ( π$_{1}$ )) = 1 / 2, the LHS expression is smaller than k$_{1}$ ‖ π$_{1}$ − π$_{2}$ ‖ . Therefore, ‖ π$_{1}$ − π$_{2}$ ‖ ≥ c/k$_{1}$k$_{2}$ . Lemma B.8. Suppose the existence of a deterministic L > 0 such that, for all m ∈ R and sufficiently small ϵ > 0 , with P -probability one, Then, for all m ∈ R and π$_{1}$, π$_{2}$ ∈ Π , | ˙ F$_{π}$$_{2}$ ( m ) − ˙ F$_{π}$$_{1}$ ( m ) | ≲ ‖ π$_{1}$ − π$_{2}$ ‖ . Proof of Lemma B.8. Set arbitrarily π$_{1}$, π$_{2}$ ∈ Π and m ∈ R . In this proof, the universal positive multiplicative constants attached to the ≲ -inequalities do not depend on π$_{1}$, π$_{2}$, m . First, observe that Second note that, for every ϵ > 0 small enough, V ( π$_{2}$ ) − V ( π$_{1}$ ) ≤ k$_{2}$ [ F$_{π}$$_{2}$ ( V ( π$_{2}$ )) − F$_{π}$$_{2}$ ( V ( π$_{1}$ ))] = k$_{2}$ [ F$_{π}$$_{1}$ ( V ( π$_{1}$ )) − F$_{π}$$_{2}$ ( V ( π$_{1}$ ))] ≤ k$_{1}$k$_{2}$ ‖ π$_{1}$ − π$_{2}$ ‖ . F$_{π}$$_{2}$ ( V ( π$_{2}$ )) − F$_{π}$$_{2}$ ( V ( π$_{1}$ )) ≥ F$_{π}$$_{2}$ ( V ( π$_{2}$ )) − F$_{π}$$_{2}$ ( V ( π$_{2}$ ) − c ) ≥ k − 1 2 c. P ( m < Y ≤ m + ϵ | X ) ≤ Lϵ. ∣ ˙ F$_{π}$$_{2}$ ( m ) − ˙ F$_{π}$$_{1}$ ( m ) ∣ = lim ϵ ↓ 0 1 ϵ | F$_{π}$$_{2}$ ( m + ϵ ) − F$_{π}$$_{2}$ ( m ) − F$_{π}$$_{1}$ ( m + ϵ ) + F$_{π}$$_{1}$ ( m ) | . ∣ F$_{π}$$_{2}$ ( m + ϵ ) − F$_{π}$$_{2}$ ( m ) − F$_{π}$$_{1}$ ( m + ϵ ) + F$_{π}$$_{1}$ ( m ) ∣ 34