Gamma-Beta algebra

October 14, 2019 — January 12, 2022

Lévy processes
stochastic processes
time series

\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\corr}{\operatorname{Corr}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\vrv}[1]{\vv{\rv{#1}}} \renewcommand{\disteq}{\stackrel{d}{=}} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]

Figure 1: Gamma (left) and Beta (right) processes.

An addendum to the gamma distribution notebook about some specialised results in Dufresne (1998). Everything here is transcribed from that paper with only light edits.

The main insight is that we can get a long way with the Mellin transform, \[ t \rightarrow \int_{\mathbb{R}} x^{t} \mu(d x) \] (Beware \(x^{t}\) is \(x^{t-1}\) in some papers.) Our results only require \(t \in \mathbb{R}\), though the general theory of Mellin transforms is formulated for complex arguments.

The distribution with Mellin transform \[ \begin{aligned} &\frac{\Gamma\left(a_{1}+t\right) \cdots \Gamma\left(a_{p}+t\right)}{\Gamma\left(a_{1}\right) \cdots \Gamma\left(a_{p}\right)} \times \frac{\Gamma\left(b_{1}\right) \cdots \Gamma\left(b_{q}\right)}{\Gamma\left(b_{1}+t\right) \cdots \Gamma\left(b_{q}+t\right)} \\ &\times \frac{\Gamma\left(c_{1}-t\right) \cdots \Gamma\left(c_{r}-t\right)}{\Gamma\left(c_{1}\right) \cdots \Gamma\left(c_{r}\right)} \times \frac{\Gamma\left(d_{1}\right) \cdots \Gamma\left(d_{s}\right)}{\Gamma\left(d_{1}-t\right) \cdots \Gamma\left(d_{s}-t\right)} \end{aligned} \] (if it exists) is denoted \[ \left(\begin{array}{ccc|ccc} a_{1} & \cdots & a_{p} & c_{1} & \cdots & c_{r} \\ b_{1} & \cdots & b_{q} & d_{1} & \cdots & d_{s} \end{array}\right) . \] The law exists, for instance, if \(p \geq q, a_{i}<b_{i}, 1 \leq i \leq q, r \geq s\), \(c_{i}<d_{i}, 1 \leq i \leq s\), in which case it is the same as the ratio of two independent variables with distributions \[ \beta_{a_{1}, b_{1}-a_{1}} \odot \cdots \odot \beta_{a_{q}, b_{q}-a_{q}} \odot \Gamma\left(a_{q+1}, 1\right) \odot \cdots \odot \Gamma\left(a_{p}, 1\right) \] and \[ \beta_{c_{1}, d_{1}-c_{1}} \odot \cdots \odot \beta_{c_{s}, d_{s}-c_{s}} \odot \Gamma\left(c_{s+1}, 1\right) \odot \cdots \odot \Gamma\left(c_{r}, 1\right) \]

There are (at least) two useful rules captured by these symbols. The first is that the parameters which appear “above” and “below” may be cancelled, for instance if \(c_{1}=d_{2}\) then the two parameters may be omitted.

The second is for the multiplication of independent variables: \[ \begin{gathered} \left(\begin{array}{ccc|ccc} a_{1} & \cdots & a_{p} & c_{1} & \cdots & c_{r} \\ b_{1} & \cdots & b_{q} & d_{1} & \cdots & d_{s} \end{array}\right) \odot\left(\begin{array}{ccc|ccc} a_{1}^{\prime} & \cdots & a_{p^{\prime}}^{\prime} & c_{1}^{\prime} & \cdots & c_{r^{\prime}}^{\prime} \\ b_{1}^{\prime} & \cdots & b_{q^{\prime}}^{\prime} & d_{1}^{\prime} & \cdots & d_{s^{\prime}}^{\prime} \end{array}\right) \\ =\left(\begin{array}{cccccc|cccccc} a_{1} & \cdots & a_{p} & a_{1}^{\prime} & \cdots & a_{p^{\prime}}^{\prime} & c_{1} & \cdots & c_{r} & c_{1}^{\prime} & \cdots & c_{r^{\prime}}^{\prime} \\ b_{1} & \cdots & b_{q} & b_{1}^{\prime} & \cdots & b_{q^{\prime}}^{\prime} & d_{1} & \cdots & d_{s} & d_{1}^{\prime} & \cdots & d_{s^{\prime}}^{\prime} \end{array}\right) . \end{gathered} \]

TBC. For now, see the nifty application in Yor (2007).

1 References

Dufresne. 1998. Algebraic Properties of Beta and Gamma Distributions, and Applications.” Advances in Applied Mathematics.
Émery, and Yor. 2004. A Parallel Between Brownian Bridges and Gamma Bridges.” Publications of the Research Institute for Mathematical Sciences.
Roychowdhury, and Kulis. 2015. Gamma Processes, Stick-Breaking, and Variational Inference.” In Artificial Intelligence and Statistics.
Yor. 2007. Some Remarkable Properties of Gamma Processes.” In Advances in Mathematical Finance. Applied and Numerical Harmonic Analysis.