Fix a dimension $d\ge2$.

  • Let $Q_d$ denote the positive quadrant of $\mathbb{R}^d$, that is, $Q_d$ is the set of points $\mathbf{x}=(x_i)_i$ in $\mathbb{R}^d$ such that $x_i>0$ for every $i$.
  • For every $\mathbf{x}$ in $Q_d$, let $|\mathbf{x}|=x_1+\ldots+x_d$.
  • Let $\Delta_d$ denote the set of points $\mathbf{x}$ in $Q_d$ such that $|\mathbf{x}|=1$.
  • For every $\mathbf{a}$ and $\mathbf{b}$ in $Q_d$, define $\mathbf{a}\cdot \mathbf{b}$ in $Q_d$ by $(\mathbf{a}\cdot \mathbf{b})_i=a_ib_i$ for every $i$.
  • For every $\mathbf{a}$ in $Q_d$, let $\mathrm{Dir}(\mathbf{a})$ denote the Dirichlet distribution of parameter $\mathbf{a}$.

The problem in a nutshell

Fix $\mathbf{a}$ and $\mathbf{b}$ in $Q_d$. Choose a random parameter $\mathbf{u}$ in $\Delta_d$ with distribution $\mathrm{Dir}(\mathbf{a})$. Then choose a random point $\mathbf{X}$ in $\Delta_d$ with distribution $\mathrm{Dir}(\mathbf{b}\cdot \mathbf{u})$.

My aim is to understand the (absolute) distribution of $\mathbf{X}$.

Some more notations

For every $\mathbf{a}=(a_i)_i$ in $Q_d$, $\mathrm{Dir}(\mathbf{a})$ is the absolutely continuous probability measure on $\Delta_d$ whose density $f(\ |\mathbf{a})$ at $\mathbf{x}$ is proportional to $x_1^{a_1-1}\cdots x_d^{a_d-1}$. More precisely, $$ f(\mathbf{x}|\mathbf{a})=\Gamma(|\mathbf{a}|)\mathbf{x}^{\mathbf{a}-1}/\Gamma(\mathbf{a}), $$ with the following shorthands: $$ \Gamma(\mathbf{a})=\Gamma(a_1)\cdots\Gamma(a_d),\quad \mathbf{x}^{\mathbf{a}-1}=x_1^{a_1-1}\cdots x_d^{a_d-1}. $$ The density $f_{\mathbf{a},\mathbf{b}}$ of the distribution of $\mathbf{X}$ is $$ f_{\mathbf{a},\mathbf{b}}(\mathbf{x})=\int_{\Delta_d} f(\mathbf{x}|\mathbf{b}\cdot \mathbf{u})f(\mathbf{u}|\mathbf{a})\mathrm{d}u_1\cdots\mathrm{d}u_{d-1}. $$

Some special cases

If $a_i=b_i=1$ for every $i$, $\displaystyle f_{\mathbf{1},\mathbf{1}}(\mathbf{x})\propto\int_{\Delta_d} \frac{\mathbf{x}^{\mathbf{u}-1}}{\Gamma(\mathbf{u})}\mathrm{d}u_1\cdots\mathrm{d}u_{d-1}.$

The case $d=2$ yields $$ f_{\mathbf{1},\mathbf{1}}(x,1-x)\propto\int_0^1\frac{x^{w-1}(1-x)^{-w}}{\Gamma(w)\Gamma(1-w)}\mathrm{d}w=\frac1{\pi x}\int_0^1\left(\frac{x}{1-x}\right)^{w}\sin(\pi w)\mathrm{d}w, $$ hence $$ f_{\mathbf{1},\mathbf{1}}(x,1-x)=\frac1{x(1-x)}\frac1{\pi^2+(\log[x/(1-x)])^2}. $$ Writing $\mathbf{X}=(X_1,X_2)$ with $X_1\ge0$, $X_2\ge0$ and $X_1+X_2=1$, this can be rewritten as the fact that, for every $x$ in $(0,1)$, $$ P(X_1\le x)=P(X_2\le x)=\frac12+\frac1\pi\arctan\left(\frac1\pi\log\left(\frac{x}{1-x}\right)\right). $$ Are there other cases where the density $f_{\mathbf{a},\mathbf{b}}$ is (reasonably) explicit? Or, for example, where the moments $E(\mathbf{X}^\mathbf{n})$ of $\mathbf{X}$ with $\mathbf{n}=(n_1,\ldots,n_d)$ any $d$-uplet of integers, are (reasonably) explicit?


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.