Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences

Theorem 1. Let $$ be an adapted sequence of 𝔹-valued random variables. Then, for any ε, γ > 0, and q ≥ 1,

Theorem 2. Let $$ be a finite sequence of 𝔹-valued martingale differences. For any ε > 0,  γ ∈ (1,2],  q ≥ 1, and L ∈ ℕ, if 𝔹 is γ-smooth, then

where C(γ, q, L) is a constant only depending on γ, q, and L.

Corollary 3. Let {(X_j, ℱ_j)} _j≥1 be a sequence of 𝔹-valued martingale differences. Suppose that, for some γ ∈ (1,2],

If 𝔹 is γ-smooth, then S_∞ converges a.s. and the inequalities (14) and (15) hold with n replaced by ∞.

Proof of Theorem 1. The first inequality is obvious. We only consider the second one. Clearly,

Since $$ is an adapted sequence of 𝔹-valued random variables, by Markov′s inequality (conditional on ℱ_j−1), Hence, by summing, we obtain Therefore, the upper bound in (19) gives a lower bound of $$ by (17), which implies the second inequality of (14).

Proof of Theorem 2. The first inequality is obvious, because if max _1≤j≤n∥S_j∥ ≤ ε, then

We will prove the second inequality. For any ε > 0, n ∈ ℕ^*, and L ∈ ℕ, Define where by convention inf ∅ = +∞. It is easily seen that T(j) are stopping times (cf. e.g., [34] for the definition) with respect to the filtration ℱ₀ = {∅, Ω} ⊂ ℱ₁ ⊂ ⋯⊂ℱ_n ⊂ ℱ_n+1 ⊂ ⋯, where we take ℱ_k = ℱ_n for all k ≥ n. As usual, we will write ℱ_T(j) = {A ∈ ℱ_n : A∩{T(j) = k} ∈ ℱ_k, 1 ≤ k ≤ n}. Notice that for 1 ≤ j ≤ L + 1, if T(j) < ∞, then ∥S_T(j) − S_T(j−1)∥ > ε/(L + 1); conversely, if there exists a positive integer i ∈ (T(j − 1), n] such that ∥S_i − S_T(j−1)∥ > ε/(L + 1), then T(j) < ∞. We proceed by three steps to estimate the second term of the right hand side of (21).

(a) We first prove that

which implies that where Assume that the first event in (23) takes place. Since $$, there exists m ∈ {1,2, …, n} such that ∥S_m∥ > 2ε. As ∥S_m∥ > 2ε, and it is clear that T(1) ≤ m < ∞.

Let M be the largest j ∈ [1, m] such that T(j) ≤ m. Then, T(M) ≤ m.

We will prove that M ≥ L + 1. Suppose that M ≤ L. Then, by the definition of M, T(M + 1) > m so that

As ∥x + y∥ ≤ ∥x∥ + ∥y∥, it follows that where the last step holds because ∥S_T(j)−1 − S_T(j−1)∥ ≤ ε/(L + 1) by the definition of T(j) and $$. As $$, by (26), (27), and the subadditivity of ∥·∥, we know that This is a contradiction with ∥S_m∥ > 2ε, which proves that M ≥ L + 1.

Therefore, T(j) < ∞ for all j ∈ [1, L + 1]. Thus, (23) holds.

(b) We next give an estimation of $$. For 0 ≤ r ≤ n,

Using successively Doob′s inequality in a real separable Banach space [37, Theorem 3.1], the inequality in Assouad [38, Proposition 2] (conditional on ℱ_r), and the subadditivity of the function x ↦ x^γ/2    (x ≥ 0), we obtain Therefore, by (29), Summing over r, we obtain

(c) We finally give un upper bound for the term of the right hand side of (24), using (32). Set

Applying (32) for j = 1, we see that Now, for any x > 0, where Notice that where (34) has been used for the last inequality. For 2 ≤ i ≤ L + 1, by (32), together with Markov′s inequality and Jensen′s inequality, we have Therefore, by (35), where

A simple calculation shows that

where $$. Therefore, Together with (21) and (24), this proves (15).

Proof of Corollary 3. By Theorem 2 with q = 1 and L = 0,

By Theorem 1 and Markov′s inequality, Therefore, which is equivalent to Since sup _j,k≥n∥S_k − S_j∥ is decreasing in n, this implies that which gives the desired conclusion.

Theorem 4. Assume that for some γ ∈ (1,2], as n → ∞,

If 𝔹 is γ-smooth, then for all ε > 0, as n → ∞,

Proof. Notice that, by Corollary 3, the condition (50) implies the a.s. convergence of S_n,∞. Equation (51) comes from (50) as

(52) follows from (51) and Theorem 1; (53) is a consequence of (52) and Theorem 2; (54) is implied by (53) and Corollary 3.

Lemma 5. Let ϕ : ℕ → [0, ∞) be a positive function. Suppose that for some γ ∈ (1,2], q ≥ 1 and some integer L ≥ 0,

If 𝔹 is γ-smooth, then the following conditions are equivalent:

Proof. Equation (57) are equivalent by Theorem 2.

Lemma 6. Let ϕ : ℕ → [0, ∞) be a positive function. Suppose that for some γ ∈ (1,2] and q ∈ [1, ∞),

Then, the following conditions are equivalent:

Proof. The conclusion comes directly from Theorem 1.

Lemma 7. Let ϕ : ℕ → [0, ∞) be a positive function. Suppose that (58) holds for some γ ∈ (1,2] and q ∈ [1, ∞). Then,

is implied by

Proof. The equivalence is an immediate consequence of Corollary 3.

Theorem 8. Let ϕ : ℕ → [0, ∞) be a positive function. Suppose that for some γ ∈ (1,2], q ∈ [1, ∞] and λ ∈ (0, q),

If 𝔹 is γ-smooth, then one has the following implications (63)⇔(64)⇔(65)⇒(66):

Remark 9. The condition (62) holds if for some γ ∈ (1,2], r ∈ ℝ and ε₁ > 0,

Proof of Theorem 8. Notice that when (62) holds for q = ∞ and some λ ∈ (0, ∞), then for q ∈ (λ, ∞),

Therefore, we can assume that (62) holds for some q ∈ [1, ∞) and λ ∈ (0, q). Since as L → ∞, q(1 + L)/(q + L) → q > λ, we can choose an integer L ≥ 0 large enough such that q(1 + L)/(q + L) > λ. Therefore, the condition (56) holds with o(1). By Lemma 6, (63) and (64) are equivalent; by Lemma 5, (64) and (65) are equivalent; by Lemma 7, (65) implies (66).

Theorem 10. Let ϕ : ℕ → [0, ∞) be a positive function. Suppose that for some γ ∈ (1,2], q ∈ [1, ∞] and λ ∈ (0, q),

If 𝔹 is γ-smooth, then one has the following implications (70)⇔(71)⇔(72)⇒(73):

Remark 11. Theorem 10 also holds if m_n(γ) is replaced by $$ for some M ≥ 1. In fact, the case M ≥ 2 can be reduced to the case M = 1 by considering the subsequences {(X_n,lM+i, ℱ_n,lM+i)} _l≥0 (1 ≤ i ≤ M) of {(X_nk, ℱ_nk)} _k≥1, which are still sequences of 𝔹-valued martingale differences.

Corollary 12. Suppose that (67) holds for some γ ∈ (1,2], r ∈ ℝ, and ε₁ > 0. Then one has the implications (70)⇔(71)⇔(72)⇒(73).

Proof of Theorem 10. As in the proof of Theorem 8, we can assume that q < ∞. Since as L → ∞, q(1 + L)/(q + L) → q > λ, we can choose an integer L ≥ 0 large enough such that q(1 + L)/(q + L) > λ. Let n₀ be large enough such that $$ for all n ≥ n₀. Then,

By Theorem 1, (70) and (71) are equivalent; by Theorem 2, (71) and (72) are equivalent; since (69) implies 𝔼m_n(γ) < ∞ for each n ≥ 1 with ϕ(n) > 0, by Corollary 3, (72) implies (73).

Proof of Corollary 12. Choose λ > (r + 1)/ε₁, then by (67), we have

So, the condition (69) holds for q = ∞, and the conclusion follows from Theorem 10.

Theorem 13. Let α ∈ ℝ. Assume that for some γ ∈ (1,2], as n → ∞,

If 𝔹 is γ-smooth, then for all ε > 0, as n → ∞,

Proof. It suffices to apply Theorem 4 for the array of 𝔹-valued martingale differences {(Y_nj, 𝒢_nj), j ≥ 1, n ≥ 1} defined by

Theorem 14. Let α ∈ ℝ and ϕ : ℕ → [0, ∞) be a positive function. Suppose that for some γ ∈ (1,2],  q ∈ [1, ∞] and λ ∈ (0, q),

If 𝔹 is γ-smooth, then one has the following implications (82)⇔(83)⇔(84)⇒(85):

Corollary 15. Let 1/2 < α ≤ 1 and b ≥ 0. Let l(·) > 0 be a function slowly varying at ∞ and ϕ(n) = n^bl(n). Suppose that for some γ ∈ (1/α, 2], q ∈ [1, ∞] with q > b/(γα − 1),

If 𝔹 is γ-smooth, then one has the implications (82)⇔(83)⇔(84)⇒(85).

Remark 16. It is obvious that (86) holds with q = ∞ if for some constant K > 0, all n ≥ 1 and j ≥ 1,

Proof of Theorem 14. It suffices to apply Theorem 8 for the array of 𝔹-valued martingale differences {(Y_nj, 𝒢_nj),  j ≥ 1,  n ≥ 1} defined by (80).

Proof of Corollary 15. Since γ > 1/α, we have

As q > b/(γα − 1), we can choose λ ∈ (b/(γα − 1), q). For this λ, Thus, the condition (81) holds, so that the conclusion follows from Theorem 14.

Theorem 17. Let α ∈ ℝ and ϕ : ℕ → [0, ∞) be a positive function. Suppose that for some γ ∈ (1,2], q ∈ [1, ∞] and λ ∈ (0, q),

If 𝔹 is γ-smooth, then one has the following implications (91)⇔(92)⇔(93)⇒(94):

Corollary 18. Let α > 1/2 and b ≥ 0. Let l(·) > 0 be a function slowly varying at ∞ and ϕ(n) = n^b−1l(n). Suppose that (86) holds for some γ ∈ (max {1/α, 1}, 2] and q ∈ [1, ∞] with q > b/(γα − 1). If 𝔹 is γ-smooth, then one has the implications (91)⇔(92)⇔(93)⇒(94).

Remark 19. As explained in Remark 11, Corollary 18 also holds if $$ is replaced by $$ for some M ≥ 1.

Proof of Theorem 17. It suffices to apply Theorem 10 for the array of 𝔹-valued martingale differences {(Y_nj, 𝒢_nj),  j ≥ 1,  n ≥ 1} defined by (80).

Proof of Corollary 18. Notice that

As q > b/(γα − 1), we can take λ ∈ (b/(γα − 1), q).

Then,

Thus, the result holds by Theorem 17.

Corollary 20. Let {(X_nj, ℱ_nj)} _1≤j≤n (n ≥ 1) be sequences of identically distributed 𝔹-valued martingale differences. Let p ∈ [1,4). Suppose that (86) holds for some γ ∈ (max {p/2, 1}, 2] and q ∈ [1, ∞] with q > p/(2γ − p). If 𝔹 is γ-smooth, then $$ if and only if

Proof of Corollary 20. It suffices to apply Corollary 18 with α = 2/p and l(n) = 1: we just need to check that in the present case, (91) is equivalent to 𝔼∥X₁₁∥^p < ∞. In fact, we have

As $$, the last expectation is finite if and only if $$.

Lemma 21. Let α > 0. Then, for any υ ≥ 1 and any ε > 0,

Let ε, α, b, δ > 0 and ϕ(v) = v^bl(v). If there exists v₀ > 0, such that for all v ≥ v₀, then there exists $$ depending only on v₀,  b, and α, such that for all $$, where C(b, α) = 2/(2^b/(2α) − 1).

Proof. The first inequality of (102) is obvious. If

then Thus, the second inequality of (102) holds.

Assume that for some v₀ > 0 and all v ≥ v₀, (103) holds (with the notation introduced in the lemma). Then, there exists υ₂ = υ₂(v₀, b, α) > 0, such that for all υ₁ ≥ υ₂,

Set β = b/α. Then, applying (103) for $$, we see that, for any υ₁ ≥ υ₂, Set $$. Since l is slowly varying at ∞, by Potter′s Theorem (cf. Theorem 1.5.6 in [39, page 25]), for A = 2 and δ₁ = b/2 > 0, there exists υ₃ = υ₃(b, α) such that for all υ ≥ υ₃, $$. Thus, there exists $$ such that for all $$,

Theorem 22. Let α, b > 0 and ϕ(n) = n^bl(n). Then, the following assertions are equivalent:

Proof. We use Lemma 21. By the second inequality of (102), we see that (112) implies (111); by the first inequality of (102), we know that (111) implies (110). As (103) implies (104), we see that (110) implies (112). Thus (110), (111), and (112) are all equivalent.

Lemma 23. Let α > −1. Then for some n₀, c₁, c₂ > 0 and all N ≥ n₀,

Proof. Without loss of generality, we suppose that l(s) has the form (101) with c(s) ≡ 1. Therefore, for δ ∈ (0, α + 1), s^δl(s) is increasing in [n₁, ∞) for some n₁ > 0 large enough. Consequently, for some positive constants c₀, c₂,   and  c₃ (which may depend on n₁) and all N ≥ n₁,

Similarly, for N ≥ 2n₁, Since l is slowly varying at ∞, by Potter′s Theorem, for A = 2 and δ₁ = 1, there exists n₂ such that for all N ≥ n₂, l(N/2) ≥ 2⁻²l(N). If δ ≤ α, then ∑_N/2<n≤N n^α−δ ≥ (N/2) ^α−δ(N − [N/2]) ≥ c₄N^α+1−δ for some positive constant c₄; if δ > α, then ∑_N/2<n≤N n^α−δ ≥ N^α−δ(N − [N/2]) ≥ c₅N^α+1−δ for some positive constant c₅. So, for some constants c₁ > 0, n₀ = max {2n₁, n₂}, and all N ≥ n₀,

Theorem 24. Let α, b > 0 and ϕ(n) = n^b−1l(n). Then, the following assertions are equivalent:

Proof. We proceed as in [34, page 394] where similar results were established for l(n) = 1 and real-valued random variables.

(a) We first prove that (117) is equivalent to

Let n₀ be large enough such that u⁻¹l(u) is decreasing in [n₀, ∞). Then, we have for n ≥ n₁ ≥ n₀ + 1 large enough. Here we have used the fact that lim _n→∞(l(n − 1)/l(n)) = 1. (To see this, notice that l(n − 1)/(n − 1) ≥ l(n)/n for n ≥ n₀ + 1, so that l(n − 1)/l(n) ≥ (n − 1)/n; in the same way, using the fact that ul(u) is increasing in $$ for $$ large enough, we obtain l(n − 1)/l(n) ≤ n/(n − 1) for all $$.) Similarly, for n ≥ n₂ ≥ n₀ + 1 large enough. Let N = max {n₁, n₂}. As $$, the conclusion that (117) is equivalent to (121) follows from (122) and (123).

(b) We next remark that (119) is equivalent to

This can be seen by the same argument as in (a).

(c) We now prove that (121) implies (124). Set β = b/α. We have

By Potter′s Theorem, for 0 < δ < b, there exists A = A(δ) > 1 such that So, Thus, (121) implies (124).

(d) We then conclude that (117), (118), and (119) are equivalent. By (a), (b), and (c), we see that (117) implies (119). By Lemma 21, we have the implications: (119)⇒(118)⇒(117).

(e) We finally prove that (119) and (120) are equivalent. We have

By Lemma 23, there exist n₀, c₁ > 0 such that By (128) and (129), we see that (119) implies By Potter′s Theorem, there exists x₀ > 0 such that when (M(ε)/ε)^1/α ≥ x₀ and (M(ε)) ^1/α ≥ x₀, Therefore (130), is equivalent to (120). Hence, (119) implies (120). A similar argument (using again Lemma 23) shows that (120) implies (119). Thus, (119) and (120) are equivalent.

Theorem 25. Let α, b > 0, l(·) > 0 be a function slowly varying at ∞ and ϕ(n) = n^bl(n). Suppose that for some γ ∈ (1,2], q ∈ [1, ∞] and λ ∈ (0, q),

where $$. If 𝔹 is γ-smooth, then one has the following implications (135)⇔(136)⇔(137)⇔(140)⇒(139)⇒(138):

Remark 26. As in Theorem 25, the conclusions of Theorem 25 remain valid if m_n(γ) is replaced by

for some M ∈ ℕ^*.

In fact, the case M ≥ 2 can be reduced to the case M = 1 by considering the subsequences {(X_lM+i, ℱ_lM+i)} _l≥0 (1 ≤ i ≤ M) of {(X_j, ℱ_j)} _j≥1, which are still sequences of 𝔹-valued martingale differences.

Corollary 27. Let 1/2 < α ≤ 1 and b > 0. Let l(·) > 0 be a function slowly varying at ∞ and ϕ(n) = n^bl(n). Suppose that for some γ ∈ (1/α, 2], q ∈ [1, ∞] with q > b/(γα − 1),

If 𝔹 is γ-smooth, then one has the implications (135)⇔(136)⇔(137)⇔(140)⇒(139)⇒(138).

Proof of Theorem 25. Applying Theorem 14 to X_nj = X_j,  ℱ_nj = ℱ_j (1 ≤ j ≤ n), and ϕ(n) = n^bl(n), we get the implications (135)⇔(136)⇔(137)⇒(138). Applying Theorem 22 to Y_n = ∥S_n∥, we know that (137) and (140) are equivalent. Obviously, we have the implications (140)⇒(139)⇒(138). Therefore, we have proved the implications (135)⇔(136)⇔(137)⇔(140)⇒(139)⇒(138).

Proof of Corollary 27. Since γ > 1/α, we have

As q > b/(γα − 1), we can choose λ ∈ (b/(γα − 1), q). For this λ, Thus, the condition (134) holds, so that the conclusion follows from Theorem 25.

Theorem 28. Let α, b > 0. Let l(·) > 0 be a function slowly varying at ∞ and ϕ(n) = n^b−1l(n). For any ε > 0, set

Suppose that for some γ ∈ (1,2], q ∈ [1, ∞] and λ ∈ (0, q), where $$. If 𝔹 is γ-smooth, then one has the following implications (153)⇔(154)⇔(155)⇔(156)⇔(148)⇔(149)⇔(150)⇒(151)⇒(152):

Remark 29. As in Theorem 25, the conclusions of Theorem 28 remain valid if m_n(γ) is replaced by

for some M ∈ ℕ^*.

Corollary 30. Let α > 1/2, b > 0. Let l(·) > 0 be a function slowly varying at ∞ and ϕ(n) = n^b−1l(n). Suppose that for some γ ∈ (max {1,1/α}, 2], q ∈ [1, ∞] with q > b/(γα − 1), (142) holds. If 𝔹 is γ-smooth, then one has the implications (153)⇔(154)⇔(155)⇔(156)⇔(148)⇔(149)⇔(150)⇒(151)⇒(152).

Proof of Theorem 28. Applying Theorem 10 to

we obtain the implications (153)⇔(156)⇔(148)⇒(152). By Theorem 24 applied to Y_n = ∥S_n∥, we see that (148), (149), and (150) are equivalent. Since $$, (149) implies (151) and (151) implies (152). Thus, we have the implications (148)⇔(149)⇔(150)⇒(151)⇒(152). Again by Theorem 24 applied to Y_n = ∥X_n∥, we know that (153), (154), and (155) are equivalent. Therefore, we have the implications (153)⇔(154)⇔(155)⇔(156)⇔(148)⇔(149)⇔(150)⇒(151)⇒(152).

Lemma 31. Let (Y_j) _j≥1 be a sequence of any 𝔹-valued random variables. If there exist n₀, K > 0, such that for some q ∈ [1, ∞] and all n ≥ n₀,

then

Proof. Let S₀ = 0, $$, n ≥ 1. Then,

As 1 − (1+1/j)^p ~ −p/j    (j → ∞), it follows that

Theorem 32. Let {(X_j, ℱ_j)} _j≥1 be 𝔹-valued martingale differences. Suppose that for some γ ∈ (1,2], there exist n₀, K > 0, such that for all n ≥ n₀,

If 𝔹 is γ-smooth, then for α > −(1 − 1/γ) and a > 1 − 1/γ,

Proof of Theorem 32. Clearly,

By Theorem 22, we see that if and only if Write $$. By the proof of Lemma 31, we have And by Theorem 2 with q = 1 and L = 0, we know that By Theorem 1, we see that By (166), we have From (173)–(175), we see that (171) holds. Thus, (170) holds, so that (168) holds.

Definition 33. For a function R_ρ regularly varying at ∞ of index ρ ≠ 0, one define, $$ as its inverse function.

Lemma 34. If R_ρ(x) = x^ρl(x) is regularly varying at ∞ of index ρ ≠ 0, where l(x) is of the canonical form $$, then its inverse function $$ is regularly varying at ∞ of index 1/ρ, where $$ is slowly varying at ∞.

Proof. Let y = R_ρ(x). Define $$. We have

We will prove that l₂(y) is slowly varying at ∞. We see that After changing variable, we have where $$. Thus, $$ is slowly varying at ∞ and so is l^*(y), which proves the desired result.

Theorem 35. Let {(X_j, ℱ_j)} _j≥1 be identically distributed 𝔹-valued martingale differences and b ≥ 0. Suppose that for some γ ∈ (1,2] and q ∈ [1, ∞] with q > b/(γ(α + 1) − 1), γ(α + 1) − 1 > 0 and γ(a − 1) + 1 > 0,

If 𝔹 is γ-smooth, then the following assertions hold.

(a) When a ≠ (b − α)/(b + 1),

if and only if

(b) When a = (b − α)/(b + 1), (183) is implied by

where $$ is defined in (181); conversely, if $$ and the function l₀ defined by (180) satisfies $$, then (183) implies (185).

Remark 36. Theorem 35 also holds if (182) is replaced by

for some γ ∈ (1,2], q ∈ [1, ∞] and λ ∈ (0, q), where

Corollary 37. Let {(X_j, ℱ_j)} _j≥1 be identically distributed 𝔹-valued martingale differences and b ≥ 0. Suppose that (182) holds for some γ ∈ (1,2] and q ∈ [1, ∞] with q > b/(γ(α + 1) − 1), γ(α + 1) − 1 > 0, and γ(a − 1) + 1 > 0. If 𝔹 is γ-smooth, then

if and only if

Corollary 38. Let {(X_j, ℱ_j)} _j≥1 be identically distributed 𝔹-valued martingale differences and b ≥ 0. Suppose that (182) holds for some γ ∈ (1,2] and q ∈ [1, ∞] with q > b/(γ(α + 1) − 1), γ(α + 1) − 1 > 0, and γ(a − 1) + 1 > 0. If 𝔹 is γ-smooth, then for β, p ∈ ℝ,

if and only if

Proof of Theorem 35. Notice that $$ is slowly varying at ∞ by Proposition  1.5.9a in [39, page 26]. Set

and ϕ(n) = n^b−1l(n) in Theorem 10, then {(X_nj, ℱ_nj), j ≥ 1, n ≥ 1} are sequences of 𝔹-valued martingale differences. For any n ≥ 1, we have By (193) and Lemma 31, we have Thus, $$ and whenever λ ∈ (b/(γ(α + 1) − 1), q). By Theorem 10, (183) is equivalent to Notice that in view of the identically distributed assumption, (196) holds if and only if By the monotonicity of the functions R_b−1(x): = x^b−1l(x) and R_a+α(x) = x^a+αl₁(x) for x > 0 large enough and the fact that R_b−1(n − 1) ~ R_b−1(n) and R_a+α(n − 1) ~ R_a+α(n) as n → ∞, it can be easily verified that the condition (197) is equivalent to Therefore, Theorem 35 is a direct consequence of the following lemma.

Lemma 39. Let b ≥ 0, α > −1, a > −α, and l(·) and l₁(·) are slowly varying at ∞. Let X₁ be any 𝔹-valued random variable. Then, the following assertions hold.

• (a)

  When a ≠ (b − α)/(b + 1), (198) holds if and only if (184) holds.

• (b)

  When a = (b − α)/(b + 1), (198) is implied by (185), where $$ is defined in (181).

Proof. We proceed as in the proof of Lemma 3.4 of Gut [20]. We distinguish three cases according to a > 1, a = 1 or a < 1. By choosing a smooth version, we can suppose that l₁ is differentiable (cf. [39]).

Case  1 (a > 1). In (198), we use the change of variables

Notice that 1 ≤ y ≤ x if and only if l₁(1) ≤ u < ∞ and $$. Therefore, (198) holds if and only if By Proposition 1.5.8 of [39, page 26], we have as u → ∞, where and hereafter a(u) ~ b(u) means that a(u)/b(u) → 1 as u → ∞. By Lemma 34, the right hand sides of (201) are regularly varying functions of index 1/(1 + α), and 1/(a + α) respectively, so that their ratio tends to ∞ as u → ∞, since 1/(1 + α) > 1/(a + α). Therefore, as u → ∞, Thus, (200) is equivalent to With the change of variable $$ and the fact that $$ (together with (179)), we see that (203) holds if and only if which is equivalent to the first condition of (184). Thus, (198) is equivalent to the first condition of (184).

Case  2 (a = 1). By the change of variables

we see that (198) holds if and only if With the change of variable $$, we know that (206) holds if and only if which is equivalent to the first condition of (184). Thus, (198) is equivalent to the first condition of (184).

Case  3 (−α < a < 1). By the change of variables

we see that (198) holds if and only if

We distinguish three cases according to a < (b − α)/(b + 1), a > (b − α)/(b + 1), or a = (b − α)/(b + 1).

(i) Suppose that a < (b − α)/(b + 1). By Proposition  1.5.8 of [39, page 26], we have as u → ∞,

so that (as in Case 1) Thus, (209) is equivalent to With the change of variable $$, we see that (212) holds if and only if which is equivalent to the second condition of (184). Thus, (198) is equivalent to the second condition of (184).

(ii) Suppose that a > (b − α)/(b + 1). By Proposition  1.5.10 of [39, page 27], we have as u → ∞,

so that as u → ∞, Thus, (209) holds if and only if (203) holds. Notice that (203) is equivalent to the first condition of (184). Thus (198) is equivalent to the first condition of (184).

(iii) Suppose that a = (b − α)/(b + 1). In this case, (209) reduces to

By Proposition  1.5.8 of [39, page 26], we have So, (209) is implied by With the change of variable $$, we see that (218) holds if and only if which is equivalent to (185). Thus, (209) is implied by (185). Therefore, (198) is implied by (185). This ends the proof of Lemma 39.

Proof of Corollary 38. We are in the case where l₁(x) = (log x) ^β (β ≥ 0). If y = R_1+α(x) = x^1+α(log x) ^β, then

Thus, Therefore, Similarly, Using (222), (223), and Theorem 35, we obtain the desired results.