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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 $\mathbf{a}$.

My

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 $$f_{\mathbf{1},\mathbf{1}}(t,1-t)\propto\int_0^1\frac{t^{w-1}(1-t)^{-w}}{\Gamma(w)\Gamma(1-w)}\mathrm{d}w=\frac1{\pi t}\int_0^1\left(\frac{t}{1-t}\right)^{w}\sin(\pi f_{\mathbf{1},\mathbf{1}}(x,1-x)\propto\int_0^1\frac{x^{w-1}(1-x)^{-w}}{\Gamma(w)\Gamma(1-w)}\mathrm{d}w=\frac1{\pi x}\int_0^1\left(\frac{x}{1-x}\right)^{w}\sin(\pi w)\mathrm{d}w,$$ hence (if I am not mistaken) $$f_{\mathbf{1},\mathbf{1}}(t,1-t)=\frac1{t(1-t)}\frac1{\pi^2+(\log[t/(1-t)])^2}f_{\mathbf{1},\mathbf{1}}(x,1-x)=\frac1{x(1-x)}\frac1{\pi^2+(\log[x/(1-x)])^2}.$$ 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 $E(\mathbf{x}^\mathbf{n})$ E(\mathbf{X}^\mathbf{n})$of$\mathbf{x}$\mathbf{X}$ with $\mathbf{n}=(n_1,\ldots,n_d)$ any $d$-uplet of integers, are (reasonably) explicit?

2 added 645 characters in body

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

My problem in a nutshell

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

My aim is to understand the (absolute) distribution of $x$.\mathbf{X}$. Some more notations For every$a=(a_i)_i$\mathbf{a}=(a_i)_i$ in $Q_d$, $\mathrm{Dir}(a)$ \mathrm{Dir}(\mathbf{a})$is the absolutely continuous probability measure on$\Delta_d$whose density$f(x|a)$f(\ |\mathbf{a})$ at $x$ \mathbf{x}$is proportional to$x_1^{a_1-1}\cdots x_d^{a_d-1}$. More precisely, $$f(x|a)=\Gamma(|a|)x^{a-1}/\Gamma(a)f(\mathbf{x}|\mathbf{a})=\Gamma(|\mathbf{a}|)\mathbf{x}^{\mathbf{a}-1}/\Gamma(\mathbf{a}),$$ with the following shorthands: $$\Gamma(a)=\Gamma(a_1)\cdots\Gamma(a_d),\quad x^{a-1}=x_1^{a_1-1}\cdots Gamma(\mathbf{a})=\Gamma(a_1)\cdots\Gamma(a_d),\quad \mathbf{x}^{\mathbf{a}-1}=x_1^{a_1-1}\cdots x_d^{a_d-1}.$$ The density$f_{a,b}$f_{\mathbf{a},\mathbf{b}}$ of the distribution of $x$ \mathbf{X}$is $$f_{a,b}(x)=\int_{\Delta_df_{\mathbf{a},\mathbf{b}}(\mathbf{x})=\int_{\Delta_d} f(x|b\cdot u)f(u|a)\mathrm{d}u_1\cdots\mathrm{d}u_{d-1}. f(\mathbf{x}|\mathbf{b}\cdot \mathbf{u})f(\mathbf{u}|\mathbf{a})\mathrm{d}u_1\cdots\mathrm{d}u_{d-1}.$$ Some special cases If$a_i=b_i=1$for every$i$,$\displaystyle f_{1,1}(x)\propto\int_{\Delta_df_{\mathbf{1},\mathbf{1}}(\mathbf{x})\propto\int_{\Delta_d} \frac{x^{u-1}}{\Gamma(u)}\mathrm{d}u_1\cdots\mathrm{d}u_{d-1}.$frac{\mathbf{x}^{\mathbf{u}-1}}{\Gamma(\mathbf{u})}\mathrm{d}u_1\cdots\mathrm{d}u_{d-1}.$