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

Christos N. Efrem

TL;DR

This work develops a unified, recursion-based framework for the joint distribution of the sum and the maximum of independent nonnegative random variables, covering both continuous and discrete cases and accommodating non-identically distributed components. The key contributions are explicit recurrence relations for the joint CDF $G_n(y,z)$ and, in the continuous case, the joint PDF $g_n(y,z)$ (and the discrete counterparts for PMFs), with clear initial conditions and proofs rooted in Leibniz's rule and the law of total probability. The paper also extends the results to i.i.d. and non-i.i.d. settings, provides detailed applications to peak-to-average ratio calculations, and offers extensions to negative values, discontinuous PDFs, and mixed variable types. These recurrences enable efficient numerical computation, serve as consistency checks for goodness-of-fit analyses, and generalize prior results to full generality across both continuous and discrete regimes.

Abstract

In this paper, the joint distribution of the sum and maximum of independent, not necessarily identically distributed, nonnegative random variables is studied for two cases: i) continuous and ii) discrete random variables. First, a recursive formula of the joint cumulative distribution function (CDF) is derived in both cases. Then, recurrence relations of the joint probability density function (PDF) and the joint probability mass function (PMF) are given in the former and the latter case, respectively. Interestingly, there is a fundamental difference between the joint PDF and PMF. The proofs are simple and mainly based on the following tools from calculus and discrete mathematics: differentiation under the integral sign (also known as Leibniz's integral rule), the law of total probability, and mathematical induction. In addition, this work generalizes previous results in the literature, and finally presents several extensions of the methodology.

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

TL;DR

This work develops a unified, recursion-based framework for the joint distribution of the sum and the maximum of independent nonnegative random variables, covering both continuous and discrete cases and accommodating non-identically distributed components. The key contributions are explicit recurrence relations for the joint CDF and, in the continuous case, the joint PDF (and the discrete counterparts for PMFs), with clear initial conditions and proofs rooted in Leibniz's rule and the law of total probability. The paper also extends the results to i.i.d. and non-i.i.d. settings, provides detailed applications to peak-to-average ratio calculations, and offers extensions to negative values, discontinuous PDFs, and mixed variable types. These recurrences enable efficient numerical computation, serve as consistency checks for goodness-of-fit analyses, and generalize prior results to full generality across both continuous and discrete regimes.

Abstract

In this paper, the joint distribution of the sum and maximum of independent, not necessarily identically distributed, nonnegative random variables is studied for two cases: i) continuous and ii) discrete random variables. First, a recursive formula of the joint cumulative distribution function (CDF) is derived in both cases. Then, recurrence relations of the joint probability density function (PDF) and the joint probability mass function (PMF) are given in the former and the latter case, respectively. Interestingly, there is a fundamental difference between the joint PDF and PMF. The proofs are simple and mainly based on the following tools from calculus and discrete mathematics: differentiation under the integral sign (also known as Leibniz's integral rule), the law of total probability, and mathematical induction. In addition, this work generalizes previous results in the literature, and finally presents several extensions of the methodology.
Paper Structure (22 sections, 18 theorems, 88 equations, 1 figure)

This paper contains 22 sections, 18 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. Main Contributions & Outline
  4. Notation
  5. Continuous Random Variables
  6. Recurrence Relation for the Joint CDF
  7. Recurrence Relation for the Joint PDF
  8. Independent and Identically Distributed Random Variables.
  9. Application: Peak-to-Average Ratio
  10. Other Applications
  11. Discrete Random Variables
  12. Recurrence Relation for the Joint CDF
  13. Recurrence Relation for the Joint PMF
  14. Independent and Identically Distributed Random Variables.
  15. Application: Peak-to-Average Ratio
  16. ...and 7 more sections

Key Result

Theorem 2.1

The joint CDF of $Y_n$ and $Z_n$, defined by equation:sum and maximum, is given by the following recurrence relation with initial condition In addition, $G_n (y,z)$ is continuous on $\mathbb{R}_+^2$ for all $n \in \mathbb{N}$.

Figures (1)

  • Figure 1: Two-dimensional lattice: each point $(l,m) \in \mathbb{Z}^2$ and each set of points $\mathcal{S} (l,m) \coloneqq \{(l',m') \in \mathbb{Z}^2 : 0 \leq l' \leq l,\, 0 \leq m' \leq m \}$ correspond to the probabilities $\widehat{g}_\tau (l,m)$ and $\widehat{G}_\tau (l,m) = \sum_{l'=0}^{l} \sum_{m'=0}^{m} {\widehat{g}_\tau (l',m')}$, respectively.

Theorems & Definitions (41)

  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • ...and 31 more