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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?
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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 $x=(x_i)_i$ \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}.$

The case $d=2$ yields $$ f_{1,1}(t,1-t)\propto\int_0^1\frac{t^{u-1}(1-t)^{-u}}{\Gamma(u)\Gamma(1-u)}\mathrm{d}u \propto\frac1t\int_0^1\left(\frac{t}{1-t}\right)^{u}\sin(\pi u)\mathrm{d}uf_{\mathbf{1},\mathbf{1}}(t,1-t)\propto\int_0^1\frac{t^{w-1}(1-t)^{-w}}{\Gamma(w)\Gamma(1-w)}\mathrm{d}w=\frac1{\pi t}\int_0^1\left(\frac{t}{1-t}\right)^{w}\sin(\pi w)\mathrm{d}w, $$ hence (if I am not mistaken) $$ f_{1,1}(t,1-t)=\frac1{t(1-t)}\frac1{\pi^2+(\log[t/(1-t)])^2}f_{\mathbf{1},\mathbf{1}}(t,1-t)=\frac1{t(1-t)}\frac1{\pi^2+(\log[t/(1-t)])^2}. $$ Are there any other cases where the density $f_{a,b}$ f_{\mathbf{a},\mathbf{b}}$ is (reasonably) explicit? Or, for example, where the moments $E(x^n)$ E(\mathbf{x}^\mathbf{n})$ of $x$ \mathbf{x}$ with $\mathbf{n}=(n_1,\ldots,n_d)$ any $d$-uplet of integers, are (reasonably) explicit?
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On random Dirichlet distributions

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

My problem in a nutshell

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

My aim is to understand the (absolute) distribution of $x$.

Some more notations

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

Some special cases

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

The case $d=2$ yields $$ f_{1,1}(t,1-t)\propto\int_0^1\frac{t^{u-1}(1-t)^{-u}}{\Gamma(u)\Gamma(1-u)}\mathrm{d}u \propto\frac1t\int_0^1\left(\frac{t}{1-t}\right)^{u}\sin(\pi u)\mathrm{d}u, $$ hence (if I am not mistaken) $$ f_{1,1}(t,1-t)=\frac1{t(1-t)}\frac1{\pi^2+(\log[t/(1-t)])^2}. $$ Are there any other cases where the density $f_{a,b}$ is (reasonably) explicit? Or where the moments $E(x^n)$ of $x$ are (reasonably) explicit?