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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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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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:
(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 \(a_1\):
using (a). Now define \(u_i^1=a_1\) for \(i=1, \dots , n\). Hence we have
Using (a) and (b), we therefore have
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 \(g_i(x)\), \(i=1, \dots , m\):
and note that
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:
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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
- Exponential variables
- Erlang density
- Gamma density
- Divided differences
- Fractional calculus
Mathematics Subject Classification