Recurrence relations for the joint distribution of the sum and maximum of independent random variables

Lemma A.1 (Leibniz’s integral rule with constant limits of integration, [16, p. 276, Theorem 7.1]).

Let f ⁢ ( x , t ) , ∂ ∂ ⁡ t ⁢ f ⁢ ( x , t ) be defined and continuous on X × T , where X ≔ [ x 1 , x 2 ] and T ≔ [ t 1 , t 2 ] are subsets of R with - ∞ < x 1 ≤ x 2 < ∞ and - ∞ < t 1 < t 2 < ∞ . Moreover, let a , b ∈ X be two constants. Then, for all t ∈ T ,

Lemma A.2 (Leibniz’s integral rule with variable limits of integration).

Let f ⁢ ( x , t ) and ∂ ∂ ⁡ t ⁢ f ⁢ ( x , t ) be defined and continuous on X × T , where X ≔ [ x 1 , x 2 ] and T ≔ [ t 1 , t 2 ] are subsets of R with - ∞ < x 1 ≤ x 2 < ∞ and - ∞ < t 1 < t 2 < ∞ . In addition, let a , b : T → X be differentiable functions on T . Then, for all t ∈ T ,

Proof.

This is a consequence of the fundamental theorem of calculus, the basic form of Leibniz’s integral rule (Lemma A.1), and the chain rule for multivariate functions. Let us define the function I ⁢ ( u , v , t ) ≔ ∫ v u f ⁢ ( x , t ) ⁢ d x for all ( u , v , t ) ∈ 𝒳 2 × 𝒯 . From the fundamental theorem of calculus, we have

while ∂ ∂ ⁡ t ⁢ I ⁢ ( u , v , t ) = ∫ v u ∂ ∂ ⁡ t ⁢ f ⁢ ( x , t ) ⁢ d x due to Lemma A.1. Now, by applying the chain rule, we obtain

for all t ∈ 𝒯 . ∎

Proposition A.3.

Let h ⁢ ( x , y , z ) , ∂ ∂ ⁡ z ⁢ h ⁢ ( x , y , z ) and ∂ 2 ∂ ⁡ y ⁢ ∂ ⁡ z ⁢ h ⁢ ( x , y , z ) be defined and continuous on S ≔ X × Y × Z , where X ≔ [ x 1 , x 2 ] , Y ≔ [ y 1 , y 2 ] , and Z ≔ [ z 1 , z 2 ] are subsets of R with - ∞ < x 1 ≤ x 2 < ∞ , - ∞ < y 1 < y 2 < ∞ , and - ∞ < z 1 < z 2 < ∞ . Furthermore, suppose that ∂ ∂ ⁡ y ⁢ h ⁢ ( x , y , z ) exists on S , and a , b : Z → X are differentiable functions on Z . Then, for all ( y , z ) ∈ Y × Z ,

(A.3)

Proof.

Firstly, we will compute the partial derivative of ∫ a ⁢ ( z ) b ⁢ ( z ) h ⁢ ( x , y , z ) ⁢ d x with respect to the variable z. By virtue of Lemma A.2 (extended to functions of three variables), we obtain

for all ( y , z ) ∈ 𝒴 × 𝒵 . Secondly, by differentiating the previous equation with respect to the variable y, the second-order mixed partial derivative of ∫ a ⁢ ( z ) b ⁢ ( z ) h ⁢ ( x , y , z ) ⁢ d x is given by

Finally, we conclude the proof by applying Lemma A.1 (extended to functions of three variables) to the last integral. ∎

Remark A.4.

Proposition A.3 remains valid if we replace the assumption that ∂ 2 ∂ ⁡ y ⁢ ∂ ⁡ z ⁢ h ⁢ ( x , y , z ) is continuous on 𝒮 with the assumption that ∫ a ⁢ ( z ) b ⁢ ( z ) ∂ 2 ∂ ⁡ y ⁢ ∂ ⁡ z ⁢ h ⁢ ( x , y , z ) ⁢ d x converges uniformly on 𝒴 × 𝒵 . This follows from the fact that Lemma A.1 still holds when the continuity of ∂ ∂ ⁡ t ⁢ f ⁢ ( x , t ) on 𝒳 × 𝒯 is replaced by the uniform convergence of ∫ a b ∂ ∂ ⁡ t ⁢ f ⁢ ( x , t ) ⁢ d x on 𝒯 ; see [25] and [24, p. 260].

Lemma A.5 (Leibniz’s integral rule with constant limits of integration – modified version [11, p. 56, Theorem 2.27]).

Let f ⁢ ( x , t ) and ∂ ∂ ⁡ t ⁢ f ⁢ ( x , t ) be defined on X × T , where X ≔ [ x 1 , x 2 ] and T ≔ [ t 1 , t 2 ] are subsets of R with - ∞ < x 1 ≤ x 2 < ∞ and - ∞ < t 1 < t 2 < ∞ . Let a , b ∈ X be two constants. Moreover, assume that:

1. the function f ⁢ ( x , t ) is integrable, i.e., ∫ a b | f ⁢ ( x , t ) | ⁢ d x < ∞ for all t ∈ 𝒯 ,

2. there exists a function ϑ ⁢ ( x ) such that | ∂ ∂ ⁡ t ⁢ f ⁢ ( x , t ) | ≤ ϑ ⁢ ( x ) for all ( x , t ) ∈ 𝒳 × 𝒯 , and ∫ a b ϑ ⁢ ( x ) ⁢ d x < ∞ .

Then (A.1) holds for all t ∈ T .

Lemma A.6 (Leibniz’s integral rule with variable limits of integration – modified version).

Let f ⁢ ( x , t ) and ∂ ∂ ⁡ t ⁢ f ⁢ ( x , t ) be defined on X × T , where X ≔ [ x 1 , x 2 ] and T ≔ [ t 1 , t 2 ] are subsets of R with - ∞ < x 1 ≤ x 2 < ∞ and - ∞ < t 1 < t 2 < ∞ . Let a , b : T → X be differentiable functions on T . In addition, assume that:

1. the function f ⁢ ( x , t ) is integrable, i.e., ∫ a ⁢ ( t ) b ⁢ ( t ) | f ⁢ ( x , t ) | ⁢ d x < ∞ for all t ∈ 𝒯 ,

2. there exists a function ϑ ⁢ ( x ) such that | ∂ ∂ ⁡ t ⁢ f ⁢ ( x , t ) | ≤ ϑ ⁢ ( x ) for all ( x , t ) ∈ 𝒳 × 𝒯 , and ∫ a ⁢ ( t ) b ⁢ ( t ) ϑ ⁢ ( x ) ⁢ d x < ∞ for all t ∈ 𝒯 .

Then (A.2) holds for almost all t ∈ T .

Proof.

This result follows from similar steps as in the proof of Lemma A.2, but exploiting the fundamental theorem of calculus for Lebesgue integrals [11, pp. 105–106, Corollary 3.33 and Theorem 3.35] (instead of the classical one) as well as Lemma A.5 (instead of Lemma A.1). In particular, the fundamental theorem of calculus for Lebesgue integrals implies that ∂ ∂ ⁡ u ⁢ I ⁢ ( u , v , t ) ⁢ = a.e. ⁢ f ⁢ ( u , t ) and ∂ ∂ ⁡ v ⁢ I ⁢ ( u , v , t ) ⁢ = a.e. - f ⁢ ( v , t ) . Therefore, (A.2) is true for almost every t ∈ 𝒯 . ∎

Proposition A.7.

Let h ⁢ ( x , y , z ) , ∂ ∂ ⁡ z ⁢ h ⁢ ( x , y , z ) , and ∂ 2 ∂ ⁡ y ⁢ ∂ ⁡ z ⁢ h ⁢ ( x , y , z ) be defined on S ≔ X × Y × Z , where X ≔ [ x 1 , x 2 ] , Y ≔ [ y 1 , y 2 ] , and Z ≔ [ z 1 , z 2 ] are subsets of R with - ∞ < x 1 ≤ x 2 < ∞ , - ∞ < y 1 < y 2 < ∞ , and - ∞ < z 1 < z 2 < ∞ . Suppose that ∂ ∂ ⁡ y ⁢ h ⁢ ( x , y , z ) exists on S , and a , b : Z → X are differentiable functions on Z . Furthermore, assume that:

1. the function h ⁢ ( x , y , z ) is integrable, i.e., ∫ a ⁢ ( z ) b ⁢ ( z ) | h ⁢ ( x , y , z ) | ⁢ d x < ∞ for all ( y , z ) ∈ 𝒴 × 𝒵 ,

2. there exists a function φ ⁢ ( x , y ) such that | ∂ ∂ ⁡ z ⁢ h ⁢ ( x , y , z ) | ≤ φ ⁢ ( x , y ) for all ( x , y , z ) ∈ 𝒮 , and ∫ a ⁢ ( z ) b ⁢ ( z ) φ ⁢ ( x , y ) ⁢ d x < ∞ for all ( y , z ) ∈ 𝒴 × 𝒵 ,

3. there is a function ϕ ⁢ ( x ) such that | ∂ 2 ∂ ⁡ y ⁢ ∂ ⁡ z ⁢ h ⁢ ( x , y , z ) | ≤ ϕ ⁢ ( x ) for all ( x , y , z ) ∈ 𝒮 , and ∫ a ⁢ ( z ) b ⁢ ( z ) ϕ ⁢ ( x ) ⁢ d x < ∞ for all z ∈ 𝒵 .

Then (A.3) holds for almost all ( y , z ) ∈ Y × Z .

Proof.

In the proof of Proposition A.3, we make the following substitutions: Lemma A.1 ↦ Lemma A.5 and Lemma A.2 ↦ Lemma A.6, while the two equations with derivatives are valid almost everywhere now. Observe that the second assumption implies that ∫ a ⁢ ( z ) b ⁢ ( z ) | ∂ ∂ ⁡ z ⁢ h ⁢ ( x , y , z ) | ⁢ d x < ∞ for all ( y , z ) ∈ 𝒴 × 𝒵 , which is required to apply Lemma A.5 in the final step. ∎

Definition B.1 ([16, p. 284]).

A Dirac sequence is a sequence of functions { δ k ⁢ ( x ) } k ∈ ℕ that satisfy three properties:

1. δ k ⁢ ( - x ) = δ k ⁢ ( x ) ≥ 0 for all k ∈ ℕ and for all x ∈ ℝ ,

2. δ k ⁢ ( x ) is continuous on ℝ and ∫ - ∞ + ∞ δ k ⁢ ( x ) ⁢ d x = 1 for all k ∈ ℕ ,

3. for any ϵ , ζ > 0 , there exists K ∈ ℕ such that ∫ - ∞ - ζ δ k ⁢ ( x ) ⁢ d x + ∫ ζ + ∞ δ k ⁢ ( x ) ⁢ d x < ϵ for all integers k ≥ K .

Proposition B.2 ([16, p. 285, Theorem 1.1]).

Suppose that f ⁢ ( x ) is a bounded and piecewise-continuous function on R , D is a compact subset of R on which f ⁢ ( x ) is continuous, and { δ k ⁢ ( x ) } k ∈ N is a Dirac sequence. Then the sequence of functions { f k ⁢ ( x ) } k ∈ N , where f k ⁢ ( x ) ≔ f * δ k = ∫ - ∞ + ∞ f ⁢ ( t ) ⁢ δ k ⁢ ( x - t ) ⁢ d t , converges to f ⁢ ( x ) uniformly on D .