APPENDIX A: PROOF OF PROPOSITION 3.1

The proof of Proposition 3.1 rests on the following lemma, which establishes uncorrelatedness of certain random variables of key interest. The result is hardly new, and variants of it are folklore. Note that we do not assume a true regression relationship to hold in this lemma.

LEMMA A.1. Let Z be an n × 1 random vector whose components are uncorrelated and have constant variance. Let W be a nonstochastic n × k matrix of full column rank and let

be the ordinary least-squares estimator obtained from regressing Z on W. Let W = [W₁ : W₂] and let

be partitioned conformably, with W₁ being of dimension n × k₁, where 0 < k₁ < k. Let

be the ordinary least-squares estimator obtained from regressing Z on W₁ only. Then both

are uncorrelated with

. Furthermore,

are uncorrelated.

Proof. The first claim is simply verified by calculating the appropriate covariances. To prove the second claim, let P_W1^⊥ = (I_n − W₁(W₁′W₁)⁻¹W₁′) be the projection on the orthogonal complement of the column space of W₁ and define W° = [W₁ : V] , where V = P_W1^⊥W₂. Let

denote the ordinary least-squares estimator obtained from regressing Z on W°. Clearly,

are uncorrelated. Note that

Multiplying this identity by P_W1^⊥ from the left yields

Because V has full column rank, we obtain

. Observing that

completes the proof. █

Proof of Proposition 3.1. Repeated application of Lemma A.1 implies mutual uncorrelatedness of

. (In case p = 0, observe that

.) In view of the Gaussianity assumption the preceding list of random variables is in fact independent. By definitions (3)–(5), the proposition is then a simple consequence of this independence result. █

APPENDIX B: AUXILIARY LEMMAS

LEMMA B.1. Let r_m, m ≥ 1, and s_m, m ≥ 1, be sequences of nonnegative real numbers such that r_m − s_m and r_m s_m converge to ∞. Then

for m → ∞.

Proof. Without loss of generality we may assume that r_m − s_m is positive for each m. Define ψ(x) = φ(x)/x, where φ is the standard normal density. Then we may write

From Gradstejn and Ryzik (1985, equation (8.254), p. 931) or Feller (1966, Lemma VII.1.2), we obtain for x > 0 that 1 − Φ(x) = ψ(x)(1 − R(x)) where |R(x)| ≤ x⁻². This implies

Furthermore,

which completes the proof. █

LEMMA B.2. Let W_m be a nonnegative random variable such that mW_m² is chi-square distributed with m degrees of freedom, m ≥ 1, and let ε ≥ 0. Then

Proof. The case ε = 0 is trivial; hence assume ε > 0. Using the moment generating function of the chi-square distribution and Markov's inequality we obtain for any 0 < t < ½

Setting t(ε) = ½(2ε + ε²)/(1 + ε)² we see that 0 < t(ε) < ½ and h(t(ε),ε) = (ε(2 + ε) − 2 log(1 + ε))/2. Because log(1 + ε) < ε, we have h(t(ε),ε) > (ε(2 + ε) − 2ε)/2 = ε²/2. █

LEMMA B.3. Let W_m, m ≥ 1, be as in the previous lemma and let τ_i,m, m ≥ 1, be a sequence of nonnegative real numbers for i = 1,…,l. Assume that the sequence τ_i,m converges to some τ_i ∈ R ∪ {∞} for m → ∞ and that τ_i,m = O(m^1/2) for each i = 1,…,l. Assume that τ_i = ∞ for i = 1,…,k where 1 ≤ k ≤ l and that τ_i ∈ R for i = k + 1,…,l. Define τ_*,m = min{τ_1,m,…,τ_k,m}. Let d_i, i = 1,…,l, be positive real numbers and 0 < ε < 1. Then there exists a real number K such that for every α satisfying 0 < α < min{1/d₁,…,1/d_k}

holds. The constant K depends only on k, d_i, i = 1,…,k, and the sequences τ_i,m, i = 1,…,k, whereas the constant implicit in the O(·) notation may depend on all sequences τ_i,m, all d_i, ε, and even on α.

Proof. Because Δ(a,b) is strictly increasing in b, and because Δ(a,b) ≤ 1, the ratio in (B.1) is bounded from above by

Observe that the expectation in the denominator of (B.2) converges to

by dominated convergence because W_m → 1 in probability. Hence, we can find a constant K⁽¹⁾ such that (B.2) is bounded from above by

Observe that τ_i,m − d_iατ_*,m → ∞ for i = 1,…,k, because d_iα < 1, and that τ_i,m, 1 ≤ i ≤ k, is positive for sufficiently large m. Hence, Lemma B.1 implies that there exists a positive constant K⁽²⁾ such that (B.3) is bounded from above (for sufficiently large m) by

where

and ψ is as in the proof of Lemma B.1. Observe that 1 − d_i(1 − ε)/τ_i,m → 1 and that |1 − d_iατ_*,m /τ_i,m| ≥ 1 − d_iα > 0 for i = 1,…,k. Hence, for sufficiently large m, we can find a positive constant K⁽⁴⁾ such that (B.4) is bounded from above by

Because τ_i,m = O(m^1/2) by assumption, we can find a positive constant L depending only on the d_i's and the sequences τ_i,m, i = 1,…,k, such that 0 ≤ d_iτ_i,mτ_*,m /m ≤ L holds. Setting K = kL, (B.5) is seen to be bounded by

Application of Lemma B.2 completes the proof. █

For the remainder of this Appendix let Z be a standard normally distributed random variable that is independent of W_m for every m ≥ 1, where W_m is as in Lemma B.2. Define

and

where τ_i,m, i = 0,…,l, l ≥ 0, are real numbers and 0 < d_i < ∞, i = 0,…,l, holds. (If l = 0, the product in (B.6) is to be interpreted as unity.)

LEMMA B.4. Assume that τ_i,m ∈ R converges to τ_i ∈ R ∪ {−∞,∞} for m → ∞, i = 0,…,l, and that τ_i,m = O(m^1/2) for each i = 1,…,l. Then

Proof. Without loss of generality we may assume that all elements τ_i,m are nonnegative. Introduce the random variables

with the convention that A_m = 1 in case l = 0. We then observe that

Because f_m* = (1 − Δ(τ_0,m,d₀))⁻¹ on the event {|Z − τ_0,m| ≥ d₀} and because τ_0,m → τ₀ ∈ R ∪ {∞} implies Δ(τ_0,m,d₀) → Δ(τ₀,d₀), the lemma is proved if we can establish that

Because Δ(·,·) is continuous on (R ∪ {−∞,∞}) × R, convergence in probability of Δ(τ_i,m,W_m d_i) to Δ(τ_i,d_i) follows from W_m → 1 in probability. Because Δ(·,·) is bounded, it follows that EA_m converges to

whereas EB_m converges to

. In case τ₁,…,τ_l are all finite or l = 0, it is easy to see that the limit of EB_m is positive, and hence (B.7) immediately follows in this case.

In case at least one τ_i, i = 1,…,l, is infinite, the limits of both EA_m and EB_m are zero, and a more delicate argument is needed to establish (B.7): Without loss of generality we may assume that τ_i = ∞ for i = 1,…,k, 1 ≤ k ≤ l, and that τ_i is finite for i = k + 1,…,l. Set τ_*,m = min{τ_1,m,…,τ_k,m}. For arbitrary but fixed 0 < ε < 1, write

where

where α satisfies 0 < α < min{1/d₁,…,1/d_k}. We first show that EA_m⁽¹⁾/EB_m → 0:

because W_m → 1 in probability and Δ(τ₀,(1 + ε)d₀) < 1. Second, observe that

Next we show that EA_m⁽ⁱ⁾/EB_m → 0 for i = 3,4. Because EA_m⁽²⁾/EB_m is bounded as shown in (B.8), it suffices to establish EA_m⁽ⁱ⁾/EA_m⁽²⁾ → 0, i = 3,4. Lemma B.3 implies that EA_m⁽³⁾/EA_m⁽²⁾ = O(exp(−m(ε²/2 − Kα))), where K is a constant independent of α. Choosing α sufficiently small establishes convergence to zero of EA_m⁽³⁾/EA_m⁽²⁾. Turning to EA_m⁽⁴⁾/EA_m⁽²⁾ observe that

where we have first made use of Δ(a,b) ≤ 1 and then of monotonicity of Δ(·,·) in its second argument. Because τ_i is finite for i = k + 1,…,l, the limit of

is positive. To prove convergence to zero of EA_m⁽⁴⁾/EA_m⁽²⁾ it hence suffices to establish that

converges to zero for m → ∞. By Lemmas B.1 and B.2 the expression in (B.10) is bounded by a constant times

Because τ_i,m = O(m^1/2) and because τ_*,m → ∞ by construction, it follows that the expression in (B.11), and hence the upper bound in (B.9), converges to zero.

Taking these results together we have now established that

Because the r.h.s. of (B.12) converges to the r.h.s. of (B.7) for ε → 0, the proof is complete. █

LEMMA B.5. Assume that τ_i,m ∈ R converges to τ_i ∈ R ∪ {−∞,∞} for m → ∞, i = 0,…,l, and that τ_i,m = O(m^1/2) for i = 1,…,l. Then

Proof. Without loss of generality we may assume that all elements τ_i,m are nonnegative. Because f_m* is identically zero on the event {|Z − τ_0,m| < d₀} and because f_m ≥ 0, it suffices to show that E(f_m1{|Z − τ_0,m| < d₀}) converges to zero. Observe that

where A_m and B_m have been defined in the proof of Lemma B.4.

Note that 1{d₀W_m ≤ |Z − τ_0,m| < d₀} converges to zero in probability because using independence of Z and W_m we have that

and the upper bound in (B.14) converges to zero because W_m → 1 in probability and because Δ(·,·) is continuous and bounded. As shown in the proof of Lemma B.4, the random variables A_m and B_m converge in probability to nonzero constants in the case where all τ₁,…,τ_l are finite or l = 0. Because A_m and B_m are bounded, it follows that (B.13) converges to zero in this case.

Next consider the case when at least one τ_i for i = 1,…,l is infinite. Without loss of generality assume that the first k of these τ_i are infinite whereas the remaining l − k are finite. Using arguments identical to the ones leading to (B.14) we may write the numerator of (B.13) as

where A_m⁽¹⁾ and A_m⁽²⁾ have been defined in the proof of Lemma B.4. In that proof, it was also shown that EA_m⁽¹⁾/EB_m → 0 and lim sup_m→∞ EA_m⁽²⁾/EB_m ≤ (1 − Δ(τ₀,(1 + ε)d₀))⁻¹ for m → ∞. It follows that the lim sup of (B.13) is bounded by

for every 0 < ε < 1. Letting ε → 0 completes the proof. █

APPENDIX C: PROOFS FOR SECTION 4

Consider a sequence θ⁽ⁿ⁾ ∈ R^P and a sequence σ⁽ⁿ⁾ of positive real numbers. For p satisfying 1 ≤ p ≤ P we will need to express

in a different form. Define τ_j,n = −n^1/2μ_n,j(j)σ⁽ⁿ⁾⁻¹ξ_n,j⁻¹ for j = p,…,P, where μ_n(j) is given by (6) with θ replaced by θ⁽ⁿ⁾. Observe that f_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾}*(·|p) and f_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾}(·|p) both are a product of φ_n,p(·) (with σ⁽ⁿ⁾ replacing σ) times a function that depends on t_p but not on t₁,…,t_p−1. Decomposing φ_n,p(t) (with σ⁽ⁿ⁾ replacing σ) into the product of the marginal density of the last component and the conditional density of the first p − 1 coordinates given the last one, the p-fold integral representing the L₁-norm in (C.1) can easily be reduced to

where we have used the representations of f_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾}*(·|p) and f_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾}(·|p) given in (12), (23), and (24), respectively, and where U⁽ⁿ⁾(p,sc_p) denotes U(p,sc_p) after replacement of σ by σ⁽ⁿ⁾. Furthermore, φ(·;(σ⁽ⁿ⁾)²ξ_n,p²) denotes the density of a normally distributed random variable with mean zero and variance (σ⁽ⁿ⁾)²ξ_n,p². Performing the substitution r = σ⁽ⁿ⁾⁻¹ξ_n,p⁻¹t_p this can be rewritten as

The following probabilistic reinterpretation of (C.2) will prove useful. Let Z be a standard normally distributed random variable and let W_n for n > P be a nonnegative random variable, such that (n − P)W_n² is chi-square distributed with n − P degrees of freedom. Furthermore Z and W_n are assumed to be independent for every n > P. Define random variables f_n and f_n* by

and

where E(·) denotes the expectation operator w.r.t. the joint distribution of Z and {W_n : n > P}. It is now easy to see that (C.2), and hence (C.1), is equal to

Observe that Ef_n = Ef_n* = 1 holds in view of the definition of Δ(·,·) and independence of Z and W_n. Consequently,

and therefore

Proof of Proposition 4.1. It suffices to show for every given p, 1 ≤ p ≤ P, that (C.1), and hence (C.3), converges to zero for n → ∞ for any sequence θ⁽ⁿ⁾ ∈ M_p and σ⁽ⁿ⁾ > 0. Because θ⁽ⁿ⁾ ∈ M_p, it follows that μ_n,j(j) = 0 for j = p + 1,…,P and hence τ_j,n = 0 for j = p + 1,…,P. To verify convergence of (C.3) to zero, it suffices to show that every subsequence (n_i) contains a further subsequence (n_i(k)) such that the convergence occurs along this subsequence (n_i(k)). Because R ∪ {−∞,∞} is compact, every subsequence contains a further subsequence along which τ_p,n converges to some τ_p ∈ R ∪ {−∞,∞}. Hence, for verifying convergence of (C.3) we may assume without loss of generality that τ_p,n converges to τ_p. Set τ_j = 0 for j = p + 1,…,P. Decompose the expectation on the r.h.s of (C.3) as follows:

Upon making the identification m = n − P, l = P − p, τ_i,m = τ_i+p,m+P, τ_i = τ_i+p, W_m = W_m+P, f_m* = f_m+P*, f_m = f_m+P, d_i = c_i+p for i = 0,…,l, and noting that τ_i,m = 0 for i = 1,…,l, an application of Lemmas B.4 and B.5 establishes that (C.3) converges to zero. █

Proof of Theorem 4.2. Again it suffices to show for every given p, 1 ≤ p ≤ P, that (C.3) converges to zero for n → ∞, where now θ⁽ⁿ⁾ is any sequence in C and σ⁽ⁿ⁾ ∈ R satisfies σ⁽ⁿ⁾ ≥ σ_* > 0. By a reasoning similar to that in the proof of Proposition 4.1, we may assume without loss of generality that τ_j,n converges to some τ_j ∈ R ∪ {−∞,∞} for every j = p,…,P. For any sequence θ⁽ⁿ⁾ ∈ C the sequence θ⁽ⁿ⁾[¬p] is bounded because of the assumptions on C maintained in the theorem. Observe that μ_n,j(j) for j = p + 1,…,P depends on θ⁽ⁿ⁾ only through θ⁽ⁿ⁾[¬p] . Hence, boundedness of μ_n,j(j) for j = p + 1,…,P follows because X′X/n → Q > 0. Because ξ_n,j converges to a positive constant and because σ⁽ⁿ⁾ ≥ σ_* > 0, the relation τ_j,n = O(n^1/2) for j = p + 1,…,P follows. Decomposing (C.3) as in the proof of Proposition 4.1, we may again apply Lemmas B.4 and B.5 to conclude convergence to zero of (C.3). █

As a point of interest we note the following: the proof of Proposition 4.1 makes use of Lemmas B.4 and B.5. However, as shown in the proof, τ_i,m = 0 for i = 1,…,l holds as a result of the assumption θ⁽ⁿ⁾ ∈ M_p, and thus this corresponds to the trivial case discussed in the proofs of Lemmas B.4 and B.5. As a consequence, the proof of Proposition 4.1 does not rely on Lemmas B.1–B.3 at all. However, these lemmas and also the full complexity of the proofs of Lemmas B.4 and B.5 are required for the proof of Theorem 4.2, because of the fact that the case θ⁽ⁿ⁾ ∉ M_p (implying the presence of unbounded τ_i,m) is possible under the assumptions of this theorem.

Proof of Proposition 4.3.

Proof of Corollary 4.5. In light of Proposition 4.3 it only remains to show that the limit ν_p of

satisfies (37) under the assumptions of part (a) and equals ±∞ under the assumptions of parts (b) and (c). Note that

, defined by (6) with θ⁽ⁿ⁾ replacing θ, is given by (39). Under the assumptions of part (a) we have θ[¬p] = 0 and θ_p = 0, which shows that ν_p is indeed given by (37). For part (b) note that θ_p ≠ 0, whereas θ[¬p] = 0. This implies divergence of

to ν_p = ∞ or ν_p = −∞ in this case. Part (c) follows because condition (38) again implies divergence of

. █

Proof of Corollary 4.6. We first prove (40) with F_n,θ,σ(t|p) replaced by F_n,θ,σ*(t|p). For given t ∈ R^p and σ > 0 choose α ≠ 0 smaller than −t_p − σξ_∞,p c_p and let ρ be a real number larger than |α|. For any given θ ∈ M_p−1, define

where e^(p) is the pth standard basis vector in R^P. By Corollary 4.5(a) the c.d.f. F_{n,θ⁽ⁿ⁾,σ}*(·|p) converges to the c.d.f. of an excised normal distribution with density (36), where ν_p = α. By definition of α we have t_p < −α − σξ_∞,p c_p, the lower endpoint of the excision interval in (36). Consequently, F_{n,θ⁽ⁿ⁾,σ}*(t|p) converges to Φ_∞,p(t)(1 − Δ(−ασ⁻¹ξ_∞,p⁻¹,c_p))⁻¹. Because θ⁽ⁿ⁾ ∈ M_p[setmn ]M_p−1, it follows from Corollary 4.5(b) that F_{∞,θ⁽ⁿ⁾,σ}(t|p) = Φ_∞,p(t). This implies

as n → ∞. Because ασ⁻¹ξ_∞,p⁻¹ is finite and c_p is positive, we see that the limit in (C.4) is positive. Observing that

because |α| < ρ, the result (40) with F_n,θ,σ*(·|p) replacing F_n,θ,σ(·|p) follows. Proposition 4.1 then immediately delivers (40). █

APPENDIX D: PROOFS FOR SECTION 5

Proof of Lemma 5.1. To prove the lemma it suffices to show for every 1 ≤ p ≤ P that

for n → ∞ for any sequence θ⁽ⁿ⁾ ∈ R^P and any sequence of positive real numbers σ⁽ⁿ⁾. Arguing similarly as at the beginning of Appendix C, it is easy to see that the L₁-norm in (D.1) can be written as

where τ_j,n has been defined in Appendix C. With random variables Z and W_n as in Appendix C, the expression in (D.2) can be rewritten as

The expression in (D.3) is bounded from above by

Using a similar subsequence argument as in the proof of Theorem 4.2, we may assume without loss of generality that τ_j,n converges to a limit τ_j ∈ R ∪ {−∞,∞} for all j. Recalling that W_n → 1 in probability and that Z and W_n are independent, it is easy to see that both indicator functions in (D.4) converge to the same limit 1{|Z − τ_p| ≥ c_p} in probability, where the latter indicator function is to be interpreted as the constant 1 in case τ_p = ±∞. Because Δ(·,·) is continuous on (R ∪ {−∞,∞}) × R, convergence of the expression inside the expectation operator in (D.4) follows. Convergence of (D.4) to zero now follows from the dominated convergence theorem, because the integrand in (D.4) is bounded by 2. █

Proof of Theorem 5.2. We prove part (b) first. It suffices to show for every 0 ≤ p ≤ P that

for n → ∞ for any sequence θ⁽ⁿ⁾ ∈ R^P and any sequence of positive real numbers σ⁽ⁿ⁾. Arguing as in the proof of Lemma 5.1 we may assume that τ_j,n converges to a limit τ_j ∈ R ∪ {−∞,∞} for all j. It is easy to see that the absolute value in (D.5) can be rewritten as

in case p ≥ 1 and as

in case p = 0. Because W_n → 1 in probability, because Δ(·,·) is continuous on (R ∪ {−∞,∞}) × R and is bounded, convergence of (D.6) and (D.7) to zero for n → ∞ follows immediately. This completes the proof of part (b).

To prove part (a) observe that, using (14), (26), and the triangle inequality, the total variation distance between G_n,θ,σ and G_n,θ,σ* can be bounded from above by the sum over p = 0,…,P of

By our conventions, this reduces to |π_n,θ,σ(0) − π_n,θ,σ*(0)| in case p = 0, which goes to zero uniformly in θ and σ as already shown. For p ≥ 1 observe that a translation of the argument of the distribution function does not affect the total variation distance and that the c.d.f.s in (D.8) are both concentrated on the same subspace. It is then easy to see that (D.8) is bounded from above by the L₁-distance between f_n,θ,σ(·|p)π_n,θ,σ(p) and f_n,θ,σ*(·|p)π_n,θ,σ*(p) w.r.t. the Lebesgue measure on that subspace. An application of Lemma 5.1 then completes the proof. █

Proof of Proposition 5.3. We prove parts (b) and (c) first. Because Δ(·,·) is continuous on (R ∪ {−∞,∞}) × R, convergence of Δ(−n^1/2μ_n,j(j)σ⁽ⁿ⁾⁻¹ξ_n,j⁻¹,c_j) to Δ(−ν_jσ⁻¹ξ_∞,j⁻¹,c_j) for j = max(1,p),…,P follows immediately. Parts (b) and (c) for π_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾}*(p) now follow from (15) and (16), observing for part (c) that under the assumptions of this part at least one factor in (15) converges to zero and the others are bounded by one. Applying Theorem 5.2(b) delivers parts (b) and (c) also for π_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾}(p). Part (a) for f_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾}*(·|p)π_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾}*(p) now follows from part (b) and Proposition 4.3(a). An application of Lemma 5.1 then completes the proof of the proposition. █

LEMMA D.1. Let H_n be a sequence of distribution functions of finite mass (not necessarily equal to one) on R^k, which converges to H in total variation distance. Let a_n ∈ R^k be a sequence with limit a ∈ R^k. Assume that H_n and H are absolutely continuous w.r.t. Lebesgue measure and that the density h of H is continuous almost everywhere. Then H_n(· + a_n) converges to H(· + a) in the total variation distance.

Proof. Observe that

It hence suffices to show that H(· + a_n) converges to H(· + a) in the total variation distance. Because the density h is continuous almost everywhere, the density h(· + a_n) of H(· + a_n) converges almost everywhere to the density h(· + a) of H(· + a). Convergence of h(· + a_n) to h(· + a) in the L₁-sense now follows from Scheffé's theorem. The proof is completed by observing that ∥H(· + a_n) − H(· + a)∥_TV ≤ ∥h(· + a_n) − h(· + a)∥₁. █

Proof of Corollary 5.4. For θ⁽ⁿ⁾ as in the corollary and 1 ≤ j ≤ P recall that

is given by (39) with j replacing p. Because θ_j and θ[¬j] are zero for j > p₀(θ), one immediately sees that

converges to υ_j given in (46) for j > p₀(θ). If p₀(θ) > 0, then for j = p₀(θ) one also sees that

converges to ν_j = ±∞, because θ_j ≠ 0 and θ[¬j] = 0. Hence the assumptions of Proposition 5.3 are satisfied for any p ≥ p₀(θ). Part (a) of the corollary is now an immediate consequence of Proposition 5.3(b). To prove part (b), expand G_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾}* as in (14) and observe that the terms for p < p₀(θ) converge to zero in total variation distance because the probabilities π_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾}*(p) vanish in the limit as shown in part (a) and because the conditional c.d.f.s have total mass one. For p ≥ p₀(θ) observe that

converges to −(Q[p : p])⁻¹Q[p : ¬p]γ[¬p] and that

. Part (b) for G_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾}* then follows from an application of Proposition 5.3 together with Lemma D.1. (Observe that for the calculation of the total variation distance between corresponding terms of G_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾}* and G_∞,θ,σ,γ, the indicator functions appearing in these corresponding terms are identical and can be ignored.) The analogous result for G_{n,θ⁽ⁿ⁾,σ⁽ⁿ⁾} now follows from Theorem 5.2(a). █

Proof of Corollary 5.5. We first prove (49) with G_n,θ,σ* replacing G_n,θ,σ. For given u ∈ R^P and σ > 0 choose α ≠ 0 such that

holds. Let ρ be a real number larger than |α| and set

where e^(P) is the Pth standard basis vector in R^P. Note that

. Because θ ∈ M_P−1 and α ≠ 0, it follows that p₀(θ⁽ⁿ⁾) = P. Hence,

in view of Corollaries 4.5 and 5.4. On the other hand, G_{n,θ⁽ⁿ⁾,σ}*(u) converges to G_{∞,θ,σ,αe^(P)}(u) given by (47) with γ = αe^(P). Because u_P + α < 0, the indicator functions in (47) are zero for p < P, and hence G_{∞,θ,σ,αe^(P)}(u) reduces to

in view of Corollaries 4.5(a) and 5.4(a). Because of (D.9), the upper integration limit u_P in (D.11) is larger than the left endpoint of the excision interval. Hence the expressions in (D.10) and (D.11) differ, showing that the limits of G_{∞,θ⁽ⁿ⁾,σ}(u) and G_{n,θ⁽ⁿ⁾,σ}*(u) are different. This proves the result for G_n,θ,σ*. The result for G_n,θ,σ now follows from Theorem 5.2(a). █

Proof of Corollary 5.6.

(a) Let p satisfy 1 ≤ p ≤ P, let ρ > 0 and σ > 0 be arbitrary, and choose an arbitrary θ ∈ M_p−1. Define

with 0 < |α| < ρ and where e^(p) denotes the pth standard basis vector in R^P. Observe that θ⁽ⁿ⁾ ∈ M_p[setmn ]M_p−1 and that

. Now, (45) (with γ = 0) implies

(b) For p satisfying 0 ≤ p < P and arbitrary ρ > 0, σ > 0, and θ ∈ M_p define

, where 0 < |α| < ρ. Observe that θ⁽ⁿ⁾ ∈ M_p+1[setmn ]M_p and that

. Applying (45) (with γ = 0) implies