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On the density for sums of independent exponential, Erlang and gamma variates

Abstract

This paper re-examines the density for sums of independent exponential, Erlang and gamma random variables. By using a divided difference perspective, the paper provides a unified approach to finding closed-form formulae for such convolutions. In particular, the divided difference perspective for sums of Erlang variates suggests a new approach to finding the density for sums of independent gamma variates using fractional calculus.

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Acknowledgements

The author would like to thank Corina Constantinescu and Wei Zhu for helpful discussions. In addition, the author is pleased to acknowledge the valuable comments and suggestions of the two anonymous reviewers.

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No financial support was provided for the conduct of the research and/or preparation of this manuscript.

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Correspondence to Edmond Levy.

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Appendix

Appendix

Proof of Proposition 6.1

We prove this in two parts:

(a) Examine the differential of \(f[a_1,a_2, \dots , a_k]\) w.r.t. \(a_1\) as a limit as follows:

$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial a_1}f[a_1,a_2, \dots , a_k]&= \lim _{h \rightarrow 0}\left( \frac{f[a_1+h, a_2, \dots , a_k]-f[a_1, a_2, \dots , a_k]}{h}\right) \\&= \lim _{h \rightarrow 0}\left( \frac{f[a_1+h, a_2, \dots , a_k]-f[a_1, a_2, \dots , a_k]}{(a_1+h)-a_1}\right) \\&=\lim _{h \rightarrow 0} f[a_1, a_2, \dots , a_k,a_1+h] \\&=f[a_1^{(2)}, a_2, \dots , a_k]. \end{aligned} \end{aligned}$$

(b) Consider the derivative of \(f[u_1^1, \dots ,u_n^1,a_2, \dots , a_k]\), \(n>1\), w.r.t. \(a_1\) and where each argument \(u_i^1\) is a function of \(a_1\):

$$\begin{aligned} \begin{aligned} \frac{\partial }{\partial a_1}f[u_1^1, \dots ,u_n^1,a_2, \dots , a_k]&= \sum _{i=1}^n \frac{\partial }{\partial u_i^1}f[u_1^1, \dots ,u_n^1,a_2, \dots , a_k]\frac{du_i^1}{da_1}\\&= \sum _{i=1}^n f[u_i^1, u_1^1, \dots ,u_n^1, a_2, \dots , a_k]\frac{du_i^1}{da_1}, \end{aligned} \end{aligned}$$

using (a). Now define \(u_i^1=a_1\) for \(i=1, \dots , n\). Hence we have

$$\begin{aligned} \frac{\partial }{\partial a_1}f[a_1^{(n)},a_2, \dots , a_k]= nf[a_1^{(n+1)}, a_2, \dots , a_k]. \end{aligned}$$

Using (a) and (b), we therefore have

$$\begin{aligned} \begin{aligned} \frac{\partial ^2}{\partial a_1^2}f[a_1, a_2, \dots , a_k]&=\frac{\partial }{\partial a_1}f[a_1^{(2)},a_2, \dots , a_k]\\&= 2f[a_1^{(3)}, a_2, \dots , a_k]. \end{aligned} \end{aligned}$$

It follows then that (b) taken with (a), the proposition is proved. See also Hildebrand (1956, Ch. 2). \(\square \)

Proof of Lemma 6.1

We recall the general Leibniz rule for m differentiable functions \(g_i(x)\), \(i=1, \dots , m\):

$$\begin{aligned} \frac{\partial ^n}{\partial x^n}\{g_1(x) \dots g_m(x)\} =\sum \limits _{\begin{array}{c} r_1+\dots +r_m=n\\ r_j \ge 0 \end{array}}\frac{n!}{r_1! \dots r_m!}g_1^{(r_1)} \dots g_m^{(r_m)}, \end{aligned}$$

and note that

$$\begin{aligned} \frac{d^{k}}{d a^{k}}\Big (\frac{1}{a^n}\Big ) =\frac{(-1)^k(n+k-1)!}{(n-1)!a^{n+k}}, \quad \text { for } n \in {\mathbb {N}}. \end{aligned}$$

The partial derivative for the divided difference \(f[a_1, \dots , a_m]\) in the lemma is then found by first invoking its Lagrange polynomial representation as follows:

$$\begin{aligned} \frac{\partial ^k}{\partial a_1^{k_1} \dots \partial a_m^{k_m}}f[a_1, \dots , a_m]= & {} \frac{\partial ^k}{\partial a_1^{k_1} \dots \partial a_m^{k_m}}\Big \{\sum _{i=1}^m f(a_i)\prod \limits _{\begin{array}{c} q=1 \\ q\ne i \end{array}}^m\frac{1}{(a_i-a_q)}\Big \} \\= & {} \sum _{i=1}^m \frac{\partial ^{k_i}}{\partial a_i^{k_i}}\Big \{f(a_i)\prod \limits _{\begin{array}{c} q=1 \\ q\ne i \end{array}}^m\frac{k_q!}{(a_i-a_q)^{k_q+1}}\Big \} \\= & {} \sum _{i=1}^m k_i! \sum \limits _{\begin{array}{c} r_1+\dots +r_m=k_i\\ r_j \ge 0 \end{array}} \frac{f^{r_i}(a_i)}{r_i!}\prod \limits _{\begin{array}{c} q=1 \\ q\ne i \end{array}}^m\frac{(-1)^{r_q}(k_q+r_q)!}{(a_i-a_q)^{k_q+r_q+1}r_q!} \\&\text { (using the general Leibniz rule)} \\= & {} \sum _{i=1}^m k_i! \sum \limits _{\begin{array}{c} r_1+\dots +r_m=k_i\\ r_j \ge 0 \end{array}} \frac{(-1)^{k_i-r_i}f^{r_i}(a_i)}{r_i!}\prod \limits _{\begin{array}{c} q=1 \\ q\ne i \end{array}}^m\frac{(k_q+r_q)!}{(a_i-a_q)^{k_q+r_q+1}r_q!}. \end{aligned}$$

\(\square \)

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Levy, E. On the density for sums of independent exponential, Erlang and gamma variates. Stat Papers (2021). https://doi.org/10.1007/s00362-021-01256-x

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Keywords

  • Convolutions
  • Exponential variables
  • Erlang density
  • Gamma density
  • Divided differences
  • Fractional calculus

Mathematics Subject Classification

  • 60E05
  • 62E10
  • 26A33