APPENDIX A: Proofs for Section 2

LEMMA A.1. Consider the linear model Y_i = X_iθ + u_i for i ≥ 1, where X_i is a nonstochastic l × k regressor matrix (l ≥ 1,k ≥ 1),θ is an unknown k × 1 parameter vector, and the errors u_i are i.i.d. normal with mean zero and covariance matrix I_l, the l × l identity matrix. For

, let P_n,θ denote the distribution of the sample (Y₁′,…,Y_n′)′. Then

(i)

, where Φ is the standard normal distribution function.

(ii) P_n,ϑn is contiguous w.r.t. P_n,θn if θ_n − ϑ_n = O(n^−1/2) and

is bounded.

Proof. (i) Because P_n,θ and P_n,ϑ are mutually absolutely continuous we can write the total variation distance as

Set

and assume first that A > 0. The log-likelihood ratio log(dP_n,θ /dP_n,ϑ) is then distributed as N(A/2,A) under P_n,θ and as N(−A/2,A) under P_n,ϑ as is easily seen. Elementary calculations then show that P_n,θ(log(dP_n,θ /dP_n,ϑ) > 0) = 1 − Φ(−A^1/2/2) and P_n,ϑ(log(dP_n,θ /dP_n,ϑ) > 0) = 1 − Φ(A^1/2/2). The result then follows for A > 0. If A = 0, clearly X_iθ = X_iϑ for i = 1,…,n holds, implying P_n,θ = P_n,ϑ. Because Φ(0) = ½, the result again follows.

(ii) Because

is bounded, any subsequence contains a further subsequence such that along this subsequence

converge to finite limits. By Le Cam's first lemma (cf. van der Vaart, 1998, Lem. 6.4) it thus suffices to show that dP_n,θn /dP_n,ϑn converges in distribution under P_n,ϑn to a random variable that is almost surely positive, provided that

. The same calculations as in the proof of part (i) show that log(dP_n,θn /dP_n,ϑn) is distributed as N(−A_n /2,A_n) under P_n,ϑn, where

. Here we use the convention that N(−A_n /2,A_n) denotes point mass at zero if A_n = 0. Hence, log(dP_n,θn /dP_n,ϑn) converges in distribution under P_n,ϑn to Z ∼ N(−A/2,A), where A = χ′Cχ. It follows from the continuous mapping theorem that dP_n,θn /dP_n,ϑn converges in distribution under P_n,ϑn to exp(Z), which is always positive. █

Proof of Theorem 2.1. We apply the results of Section 3 with

, β = θ, M the real line,

. We first compute the oscillation of

over B_n. It is easy to see that the risk is a strictly increasing and continuous function of θ′θ. Hence, the oscillation is given by

, which equals 2Δ(ρ), where Δ was defined prior to Theorem 2.1. Note that Δ(ρ) is positive. Applying Lemma A.1(ii) with X_i = I_k shows that P_n,θn is contiguous w.r.t. P_n,0 for every sequence θ_n ∈ B_n. Lemma 3.1(i) then gives (3). From Lemma A.1(i) we obtain

because

. Lemma 3.2 together with Remark B.2 now implies that the l.h.s. of (4) is not less than 1 − Φ(ρ) for every 0 < ρ < ∞ satisfying δ < Δ(ρ), or equivalently Δ⁻¹(δ) < ρ. Because the l.h.s. of (4) is nonincreasing and the lower bound 1 − Φ(ρ) is increasing to 1 − Φ(Δ⁻¹(δ)) when ρ decreases to Δ⁻¹(δ), the result (4) follows. The last claim follows immediately from the first inequality in (4) observing that 1 − Φ(Δ⁻¹(δ)) → ½ for δ → 0. █

LEMMA A.2. Let F be a c.d.f. on

, k > 1, and let U be an open subset of

. Then F is continuous on U if and only if for every t = (t₁,…,t_k)′ ∈ U and every i, 1 ≤ i ≤ k, the map u_i [map ] F(t₁,…,t_i−1,u_i,t_i+1,…,t_k) is continuous at u_i = t_i.

Proof. One direction is trivial. To prove the other one, fix t ∈ U and let

denote a sequence converging to t. For sufficiently small ε_i > 0 we have Π_i=1^k[t_i − ε_i, t_i + ε_i] ⊆ U and

for n ≥ N(ε₁,…,ε_k). Hence also lim sup_n F(t(n)) and lim inf_n F(t(n)) are sandwiched between these bounds. Because (t₁ − ε₁,…,t_k − ε_k) ∈ U, it follows that F(t₁ − ε₁,…,t_k−1 − ε_k−1,t_k − ε_k) converges to F(t₁ − ε₁,…,t_k−1 − ε_k−1,t_k) for ε_k → 0. Similarly, F(t₁ + ε₁,…,t_k−1 + ε_k−1,t_k + ε_k) converges to F(t₁ + ε₁,…,t_k−1 + ε_k−1,t_k) for ε_k → 0. This shows that lim sup_n F(t(n)) and lim inf_n F(t(n)) are both sandwiched between F(t₁ − ε₁,…,t_k−1 − ε_k−1,t_k) and F(t₁ + ε₁,…,t_k−1 + ε_k−1,t_k). Observing that the arguments of the latter two functions are elements of U, we may repeat the argument until we arrive at the conclusion that F(t) ≤ lim inf_n F(t(n)) ≤ lim sup_n F(t(n)) ≤ F(t). █

LEMMA A.3. Let θ, ν, and t be elements of

and let I(θ) denote the set {i : θ_i = 0}. Then the c.d.f. F_∞,θ,ν given by (6) is continuous at t provided that t_i ≠ − ν_i holds for all i ∈ I(θ). In particular, F_∞,θ,ν is continuous on all of

if θ has only nonzero components; furthermore, F_∞,θ,ν is continuous at t for all θ if t_i ≠ −ν_i holds for all i, 1 ≤ i ≤ k.

Proof. We only need to prove the first claim. The case k = 1 and also the case k > 1 and I(θ) = Ø follow immediately from (8)–(10) and the attending discussion. Hence assume k > 1 and I(θ) ≠ Ø in the sequel. Choose an open neighborhood U of t such that any s ∈ U also satisfies s_i ≠ −ν_i for i ∈ I(θ). Assume now that F_∞,θ,ν is not continuous at t. Lemma A.2 then implies the existence of an element s* ∈ U and of an index m, 1 ≤ m ≤ k, such that the function taking s_m into F_∞,θ,ν(s₁*,…,s_m−1*,s_m,s_m+1*,…,s_k*) is not continuous at s_m = s_m*. Because this function is cadlag, it follows that it has a jump of positive height at s_m = s_m*. Hence, the event A, that T_m equals s_m*, has positive probability, where T_m denotes the mth coordinate of

. We can now find a subset J ⊆ I(θ), J possibly empty, such that the event B = B(J) defined as the intersection of A with the event {T_i = −ν_i for i ∈ J and T_i ≠ −ν_i for i ∈ I(θ)[setmn ]J} has positive probability. By possibly reducing B further, we can even achieve that the sign of T_i + ν_i is constant over B for all i ∈ I(θ)[setmn ]J, without losing the property that B has positive probability. Let u[J] denote the vector obtained from u by deleting the coordinates for i ∈ J. Note that m ∉ J because s_m* ≠ −ν_m by construction and because s_m* equals T_m on the event B. Let V_θ,ν⁰(u[J]) denote the function resulting from V_θ,ν(u) after replacing u_i with −ν_i for i ∈ J. Observe that on the event B the function V_θ,ν⁰(u[J]) is a differentiable function in a neighborhood of its argmin, the coordinates of which coincide with the coordinates of

for i ∉ J by construction. Hence, the first derivative of V_θ,ν^o(u[J]) must be zero at its argmin. This leads to a system of equations of the form

for i ∉ J, where W_i is the ith coordinate of the random vector W appearing in (7) and c_i is a constant on B depending only on Q, λ, ν, and θ. Now on B we have m ∉ J and T_m = s_m*, a fixed constant, implying that the distribution of (T_j : j ∉ J) puts positive mass on a linear space of dimension k − card(J) − 1. Together with the preceding system of equations, this contradicts the fact that the distribution of (W_i : i ∉ J) is a nonsingular multivariate normal of dimension k − card(J). █

LEMMA A.4. Suppose

is given and γ ∈ G satisfies γ_i ≠ 0 for all i ∈ I(θ), where

and I(θ) = {i : θ_i = 0}. Then for any

we have

where

is given by χ_i = θ_i for i ∉ I(θ) and χ_i = γ_i for i ∈ I(θ) and sign(χ) denotes the vector with ith coordinate equal to sign(χ_i).

Proof. From (6) and (7) one sees that F_∞,θ,ργ(t) can equivalently be expressed as

, where

For ρ → ∞ the random function V_θ,ργ*(u) converges for every

and every value of W to

which is nothing else than V_χ,0(u). In view of Geyer (1996), it follows that F_∞,θ,ργ([bull ]) converges weakly to the c.d.f. of

for ρ → ∞, which is given by Φ_Q⁻¹([bull ] + λQ⁻¹ sign(χ)/2). Because any

is a continuity point of the latter distribution, the lemma follows. █

Proof of Theorem 2.2. We again apply the results of Section 3 and use the following identifications:

is the real line,

, where ρ > 0 will be chosen later, and φ_∞,0(γ) = F_∞,0,ργ(t) with γ ∈ G, the open unit ball in

. To obtain information on the oscillation of F_n,θ(t) over B_n, we wish to apply Lemma 3.5(i) with α = 0, ζ_n = ρn^−1/2, and G₀ = {e/(2k^1/2) : e ∈ {−1,1}^k}. Observe that lim_n→∞ F_{n,ργn^−1/2}(t) = F_∞,0,ργ(t) for all γ ∈ G₀ and all ρ > (2k^1/2)max{|t_i| : 1 ≤ i ≤ k}. This follows from the weak convergence of the finite-sample c.d.f. and the fact that F_∞,0,ργ is continuous at t by Lemma A.3 (applied with θ = 0) whenever ρ satisfies ±ρ/(2k^1/2) ≠ t_i for all i. Hence the general assumptions of Lemma 3.5 are satisfied for any such choice of ρ. The oscillation of F_∞,0,ργ(t) over G₀ now satisfies

by Lemma A.4. Lemma 3.5(i) now implies that

holds. This further implies that for every δ < Δ(t) we can find a ρ large enough (depending on t, δ, Q, and λ), such that δ < lim inf_n→∞ ω(F_n,θ(t),B_n)/2 holds and that the r.h.s. is positive (because Δ(t) is so by assumption). Now fix this δ and ρ (and hence B_n). Because Lemma A.1(ii) with l = 1 implies that P_n,θn is contiguous w.r.t. P_n,0 for any sequence θ_n ∈ B_n, we may apply Lemma 3.1(i) to obtain (12). From Lemma A.1(i) we also obtain

Lemma 3.2 now implies (13). Finally, (A.1) shows that we can find a ρ large enough such that lim inf_n→∞ ω(F_n,θ(t),B_n) is positive. Lemma A.1(i) shows that the asymptotic uniform equicontinuity condition of Corollary 3.4 is satisfied for B_n. Corollary 3.4(i) now implies (14). █

Proof of Theorem 2.3. We again apply the results of Section 3 using the same identifications as in the proof of Theorem 2.2, except that now ρ satisfies ρ > ∥t∥. We now wish to apply Lemma 3.5(ii) with α = 0, ζ_n = ρn^−1/2, and G₀ = G[setmn ]{−t/ρ}. Observe that lim_n→∞ F_{n,ργn^−1/2}(t) = F_∞,0,ργ(t) for all γ ∈ G₀ by weak convergence and the fact that F_∞,0,ργ is continuous at t as long as t ≠ −ργ; cf. (9)–(11). This shows that φ_n(α + γζ_n) = F_{n,ργn^−1/2}(t) converges to φ_∞,0(γ) = F_∞,0,ργ(t) for all γ ∈ G₀. Now (9)–(11) furthermore show that F_∞,0,ργ(t) equals Φ_Q⁻¹(t + λQ⁻¹ι/2) for γ > −t/ρ and equals Φ_Q⁻¹(t − λQ⁻¹ι/2) for γ < −t/ρ, where the relation < is to be interpreted coordinatewise. Define δ_* = Φ_Q⁻¹(t + λQ⁻¹ι/2) − Φ_Q⁻¹(t − λQ⁻¹ι/2) and note that δ_* is positive. Because −t/ρ ∈ G, which is open, it now follows that for every ε > 0 we can find γ_i(ε) ∈ G₀, i = 1,2, such that |γ₁(ε) − γ₂(ε)| < ε and |φ_∞,0(γ₁(ε)) − φ_∞,0(γ₂(ε))| ≥ δ_* hold; hence the assumptions of Lemma 3.5(ii) are satisfied. Applying Corollary 3.4(ii) and observing that Lemma A.1(i) implies the asymptotic uniform equicontinuity condition in Corollary 3.4 completes the proof of (15). That (12) holds for δ < Δ(t) and ρ > ∥t∥ follows from Lemma 3.1(i) and Lemma 3.5(i) because ω(φ_∞,0,G₀) ≥ δ_* > 0 holds for ρ > ∥t∥. █

Proof of Theorem 2.4. We again apply the results of Section 3 and use the identifications

the real line,

. We first prove (17). For given ρ > |t| and n ≥ 1, the sets C_n(ε) = {−n^−1/2t,−n^−1/2(t + ε)} are subsets of B_n = {θ : |θ| < ρn^−1/2} provided, e.g., 0 < ε < (ρ − |t|)/2 holds. From the formula for G_n,θ given in Section 2.3 we obtain that the oscillation satisfies

whenever ε also satisfies ε < c_n. Furthermore, Lemma A.1 implies that the total variation distance between the measures P_n,θ, θ ∈ C_n(ε), is 2Φ(ε/2) − 1. Applying the finite-sample version of Lemma 3.2 given in Remark B.2 in Appendix B with A = {n} for fixed n gives

for all δ < |Φ(c_n + t) − Φ(−c_n + t + ε)|/2 and for all 0 < ε < min{c_n,(ρ − |t|)/2}. Observing that C_n(ε) ⊆ B_n holds and letting ε go to zero establishes (17), from which we immediately obtain (18) by letting n go to infinity.

To prove (16), we apply Lemma 3.1 with

. The contiguity condition in that lemma follows immediately from Lemma A.1(ii). Choose a sequence ε_n satisfying 0 < ε_n < min{c_n,(ρ − |t|)/2}. Then using (A.2) we obtain

which completes the proof of (16). █

APPENDIX B: Proofs for Section 3

Proof of Lemma 3.1. (i) By definition of δ*, we can find sequences β_n⁽¹⁾ ∈ B_n and β_n⁽²⁾ ∈ B_n such that lim inf_n→∞ d(φ_n(β_n⁽¹⁾),φ_n(β_n⁽²⁾)) ≥ δ*. Fix a δ < δ*/2. Because

it follows for any δ′ satisfying δ < δ′ < δ*/2 that at least one of the terms on the r.h.s. of (B.1) exceeds δ′ for sufficiently large n. Thus we can find a sequence β_n ∈ B_n (in fact β_n ∈ {β_n⁽¹⁾,β_n⁽²⁾}) such that lim inf_n→∞ d(φ_n(β_n),φ_n(α)) ≥ δ′. For any δ′′ satisfying δ < δ′′ < δ′ we have

By construction, the l.h.s. of (B.2) converges to one. By contiguity, by the assumption on

, and because δ′′ − δ > 0, the second term on the r.h.s. of (B.2) converges to zero. Hence, the first term on the r.h.s. of (B.2) converges to one, which completes the proof of (21). Assume now that (22) is not true. Then we can find a sequence of estimators

and a subsequence n(i) → ∞ such that

It follows that

along this subsequence. Observe that the statistical experiment corresponding to the subsequence n(i) and the estimators

satisfies the requirements of Lemma 3.1(i). Applying the already established result (21) to this experiment we conclude that

, which leads to a contradiction.

(ii) It suffices to show that

for every δ > 0 and every sequence β_n ∈ B_n. Because d(φ_n(β_n),φ_n(α)) ≤ ω(φ_n,B_n) holds, this follows from the assumption on

, from contiguity, and from

Remark B.1 (Extensions of Lemma 3.1). (i) If lim sup_n→∞ ω(φ_n,B_n) > 0 but lim inf_n→∞ ω(φ_n,B_n) = 0 holds, applying Lemma 3.1 to appropriate subsequences shows that (21) and (22) hold for δ < lim sup_n→∞ ω(φ_n,B_n)/2 provided the limit inferior in (21) and (22) is replaced by a limit superior; similarly, (23) holds for every δ > 0 provided the limit superior is replaced by a limit inferior. In particular, it follows that the condition lim sup_n→∞ ω(φ_n,B_n) = 0 is not only sufficient but also necessary for (23) to hold for every δ > 0. (ii) Inspection of the proof shows that Lemma 3.1 continues to hold if α is replaced by α_n ∈ B_n at every occurrence in the formulation of the lemma.

Proof of Lemma 3.2. Again we can find sequences β_n⁽¹⁾ ∈ B_n and β_n⁽²⁾ ∈ B_n such that lim inf_n→∞ d(φ_n(β_n⁽¹⁾),φ_n(β_n⁽²⁾)) ≥ δ*. For every given δ < δ*/2 and for every estimator

we then have

for an appropriate choice β_n ∈ {β_n⁽¹⁾,β_n⁽²⁾} ⊆ B_n, the choice possibly depending on

. Now for every ε > 0 there exists n₀ = n₀(δ,ε) such that for n ≥ n₀ the leftmost side of (B.3) equals one and ∥P_{n,βn⁽¹⁾} − P_{n,βn⁽²⁾}∥_TV < Γ + ε. Hence,

holds for n ≥ n₀ and every estimator

. It follows that

holds for n ≥ n₀, which implies (25). █

Remark B.2 (Finite-sample version of Lemma 3.2). Lemma 3.2 is at its core actually a finite-sample result. Let A be a nonempty subset of

. (Of primary interest is the case

or the case where A contains only one element.) If the limit superior in (24) is replaced by the supremum over A and if the limit inferior in the definition of δ* is replaced by the infimum over A, then (25) holds with the limit inferior replaced by the infimum over A. (This follows either from inspection of the proof of the lemma or from applying the lemma for every m ∈ A to φ_n′ and {P_n,β′ : β ∈ B_n′} given by φ_n′ = φ_m, P_n,β′ = P_m,β, and B_n′ = B_m for all n ≥ 1.)

LEMMA B.1. Let B_n, n ≥ 1, be a sequence of (nonempty) subsets of B. For every k ≥ 1, n ≥ 1, let C_n,k be a (nonempty) subset of B_n. Define

and assume that Γ_k < 1 holds for every k. Suppose δ_k* = lim inf_n→∞ ω(φ_n,C_n,k) is positive for every k. Then

holds. If, additionally, δ** = inf_k δ_k* is positive, then even

holds for every δ < δ**/2. In particular, if inf_k Γ_k = 0, then the lower bound in (B.5) and (B.6) reduces to ½.

Proof. Applying Lemma 3.2 to C_n,k and noting that C_n,k ⊆ B_n we obtain

for every δ < δ_k*/2 and for every k ≥ 1. Inequality (B.6) is then an immediate consequence, and (B.5) follows because the l.h.s. of the preceding inequality is nonincreasing in δ. █

Remark B.3 (Finite-sample version of Lemma B.1). Let A be a nonempty subset of

. If the limit superior in (B.4) is replaced by the supremum over A and if the limit inferior in the definition of δ_k* is replaced by the infimum over A, then (B.5) and (B.6) hold with the limit inferior replaced by the infimum over A.

Proof of Corollary 3.3. Because α is interior to B_n, we may without loss of generality set B_n = {β ∈ B : π(α,β) < ζ_n} for sufficiently small ζ_n that are positive and decrease to zero. By further reducing ζ_n if necessary, we may—in view of the asymptotic continuity of the maps β [map ] P_n,β at α and the triangle inequality—conclude that (24) holds with Γ = 0. Because ω(φ,B_n) ≥ η clearly holds, δ* defined in Lemma 3.2 is not less than η, which is positive as a result of discontinuity of φ at α. An application of Lemma 3.2 then completes the proof. █

Proof of Corollary 3.4. (i) We first construct, for every k ≥ 1, sequences β_n,1^(k) and β_n,2^(k) in B_n satisfying

for every n ≥ 1 and

By definition of δ*, we can obviously find β_n,1⁽¹⁾ and β_n,2⁽¹⁾ in B_n such that (B.8) holds for k = 1. The triangle inequality applied to β_n,1⁽¹⁾,β_n,2⁽¹⁾, and α shows that also (B.7) is satisfied for k = 1. For k > 1 define χ_n^(k)(i) = β_n,1⁽¹⁾ + (i/k)(β_n,2⁽¹⁾ − β_n,1⁽¹⁾), 0 ≤ i ≤ k, and observe that χ_n^(k)(i) ∈ B_n by convexity of B_n. Because

there exists an index i_n, 0 ≤ i_n < k, such that

Define β_n,1^(k) = χ_n^(k)(i_n) and β_n,2^(k) = χ_n^(k)(i_n + 1). Relation (B.8) immediately follows, and (B.7) follows because ∥β_n,1^(k) − β_n,2^(k)∥ ≤ ∥β_n,1⁽¹⁾ − β_n,2⁽¹⁾∥/k holds by construction.

Define the sets C_n,k = {β_n,1^(k),β_n,2^(k)} and apply Lemma B.1. Clearly, δ_k* = lim inf_n→∞ ω(φ_n,C_n,k) is not less than δ*/k and hence is positive for every k ≥ 1. Furthermore, Γ_k reduces to lim sup_n→∞∥P_n,βn,1^(k) − P_n,βn,2^(k)∥_TV, which converges to zero for k → ∞ in view of (B.7) and the asymptotic uniform equicontinuity assumption. Hence Γ_k < 1 holds at least for k ≥ k₀. The result now follows from (B.5). The final claim follows from (B.6) upon observing that now δ** = δ_k* = ∞.

(ii) Define C_n,k = {β_n,1(k⁻¹),β_n,2(k⁻¹)} and apply Lemma B.1. Observe that δ_k* ≥ δ_* > 0, and hence δ** ≥ δ_*, holds in view of (29). Furthermore, Γ_k converges to zero for k → ∞ in view of (28) and the asymptotic uniform equicontinuity assumption. The result then follows from (B.6). █

Remark B.4 (Finite-sample versions of Corollaries 3.3 and 3.4). Let A be a nonempty subset of

. (Of primary interest is the case

or the case where A contains only one element.)

(i) Suppose the asymptotic continuity condition in Corollary 3.3 is replaced by continuity of the maps β [map ] P_n,β at α for every n ∈ A. Then (26) continues to hold provided the limit inferior is replaced by the infimum over n ∈ A. (This follows from Lemma B.1 together with Remark B.3 upon observing that the assumed continuity property of β [map ] P_n,β allows one to choose open balls C_n,k ⊆ B_n centered at α such that Γ_k → 0 for k → ∞, and from the fact that ω(φ,C_n,k) ≥ η > 0 holds for n ∈ A and all k ≥ 1.)

(ii) Suppose the asymptotic uniform equicontinuity condition in Corollary 3.4 is replaced by the condition

Then (i) and (ii) of that corollary continue to hold if at every occurrence a limit inferior (superior) is replaced by an infimum (supremum) over n ∈ A. The proof is similar to the proof of Corollary 3.4 using Remark B.3 instead of Lemma B.1 (and replacing δ* on the r.h.s. of (B.8) by some δ⁺, 0 < δ⁺ < δ*).

Proof of Lemma 3.5. (i) For every ω′ < ω(φ_∞,α,G₀) we can find γ₁, γ₂ in G₀ such that d(φ_∞,α(γ₁),φ_∞,α(γ₂)) > ω′. From pointwise convergence it follows that δ* ≥ ω′ and hence δ* ≥ ω(φ_∞,α,G₀) because ω′ has been chosen arbitrarily, subject to ω′ < ω(φ_∞,α,G₀). The oscillation ω(φ_∞,α,G₀) is positive, because φ_∞,α is nonconstant on G₀.

(ii) Define β_n,i(ε) = α + γ_i(ε)ζ_n. Then (28) and (29) are obviously satisfied.

(iii) Discontinuity implies that η_∞ is positive. Now, for every 0 < δ_* < η_∞ we can find γ₁(ε), γ₂(ε) in G₀ (near γ₀) satisfying the assumptions of the already established part (ii) of the lemma.

(iv) Follows trivially from uniform convergence. █

Proof of Lemma 3.6. (i) Assume P_n,αn*(F_n*) → 0. This can be rewritten as E_n,αn(f_n) → 0, where f_n is given by f_n(ω) = ∫1_Fn*(ω,ξ)K_n(ω,dξ) and E_n,αn denotes expectation w.r.t. P_n,αn. Clearly, 0 ≤ f_n ≤ 1 holds. For every ε > 0 we have E_n,αn(f_n) ≥ εP_n,αn(f_n ≥ ε), implying P_n,αn(f_n ≥ ε) → 0. By contiguity also P_n,βn(f_n ≥ ε) → 0. Because of 0 ≤ E_n,βn(f_n) ≤ ε + P_n,βn(f_n ≥ ε), we obtain lim sup_n→∞ E_n,βn(f_n) ≤ ε. But this proves P_n,βn*(F_n*) = E_n,βn(f_n) → 0.

(ii) By specializing to the events F × Ξ_n with

, one immediately sees that the l.h.s. in (ii) is not less than the r.h.s. The reverse inequality is proved, e.g., in Pötscher (2002), Remark 2.1. █

APPENDIX C: Conditional Experiments

In this Appendix we show how the results in Section 3 can be easily transferred from a given statistical experiment to a derived experiment obtained by conditioning. We start with the following lemma.

LEMMA C.1.

(i) Let Q and R be probability measures defined on some measurable space

and let

satisfy Q(E) > 0 and R(E) > 0. Then ∥Q(·|E) − R(·|E)∥_TV ≤ 2∥Q − R∥_TV /max(Q(E),R(E)).

(ii) For n ≥ 1, let Q_n and R_n be probability measures defined on measurable spaces

and let

satisfy Q_n(E_n) > 0 and R_n(E_n) > 0 for each n ≥ 1. Suppose lim inf_n→∞ R_n(E_n) > 0 holds. If R_n is contiguous w.r.t. Q_n, then R_n(·|E_n) is contiguous w.r.t. Q_n(·|E_n).

Proof. (i) For arbitrary

we have |Q(A|E) − R(A|E)| = |R(E)Q(A ∩ E) − Q(E)Q(A ∩ E) + Q(E)Q(A ∩ E) − Q(E)R(A ∩ E)|/(Q(E)R(E)) ≤ |Q(E) − R(E)|Q(A ∩ E)/(Q(E)R(E)) + |Q(A ∩ E) − R(A ∩ E)|/R(E) ≤ 2∥Q − R∥_TV /R(E). Reversing the rôles of Q and R we obtain |Q(A|E) − R(A|E)| ≤ 2∥Q − R∥_TV /Q(E).

(ii) If Q_n(F_n|E_n) → 0 then Q_n(F_n ∩ E_n) → 0. Contiguity gives R_n(F_n ∩ E_n) → 0. Because lim inf_n→∞ R_n(E_n) > 0 holds, R_n(F_n|E_n) → 0 follows. █

For the rest of this Appendix we use the notation of Section 3. In particular, let B_n, n ≥ 1, denote a sequence of (nonempty) subsets of B and let

satisfy P_n,β(E_n) > 0 for every β ∈ B_n and n ≥ 1. The first corollary follows immediately from Lemma C.1.

COROLLARY C.2. Suppose

holds.

A simple sufficient condition for (C.1) is that lim inf_n→∞ P_n,αn(E_n) > 0 for some sequence α_n ∈ B_n and that P_n,αn is contiguous w.r.t. P_n,βn for all sequences β_n ∈ B_n.

COROLLARY C.3. Let α_n ∈ B_n, n ≥ 1, be a given sequence and assume that m = lim inf_n→∞ P_n,αn(E_n) > 0 holds. If {P_n,β : β ∈ B_n} satisfies (24), then {P_n,β(·|E_n) : β ∈ B_n} satisfies (24) with Γ replaced by 4Γ/m.

Proof. Applying the triangle inequality and Lemma C.1 we obtain for β, γ ∈ B_n

Taking the supremum over B_n × B_n followed by the limsup leads to the bound 4Γ/m. █

Remark C.1. The asymptotic continuity condition in Corollary 3.3 immediately carries over from the measures P_n,β to the conditional measures P_n,β(·|E_n) provided lim inf_n→∞ P_n,α(E_n) > 0 holds. The same is true for the asymptotic uniform equicontinuity condition in Corollary 3.4 provided condition (C.1) is satisfied.