Merge pull request #904 from Franzi2114/patch-1

Update positive_lower-bounded_distributions.qmd; correction wiener description. Closes #748

Author: avehtari

src/functions-reference/positive_lower-bounded_distributions.qmd

@@ -198,14 +198,22 @@ f_s(t^*\mid0,1,\beta) = \sum_{k=-\infty}^\infty \frac{1}{\sqrt{2\pi(t^*)^3}}(\be
 
 Which of these is used in the computations depends on which expression requires the smaller number of components $k$ to guarantee a pre-specified precision
 
-In the case where $s_{\delta}$, $s_{\beta}$, and $s_{\tau}$ are all $0$, this simplifies to
+In the case where $s_{\delta}$, $s_{\beta}$, and $s_{\tau}$ are all $0$, this simplifies to one representation that converges fast for small reaction-time values ("small time expansion"):
 \begin{equation*}
 \text{Wiener}(y|\alpha, \tau, \beta, \delta) =
-\frac{\alpha^3}{(y-\tau)^{3/2}} \exp \! \left(- \delta \alpha \beta -
+\frac{\alpha}{(y-\tau)^{3/2}} \exp \! \left(- \delta \alpha \beta -
 \frac{\delta^2(y-\tau)}{2}\right) \sum_{k = - \infty}^{\infty} (2k +
-\beta) \phi \! \left(\frac{2k \alpha + \beta}{\sqrt{y - \tau}}\right)
-\end{equation*} where $\phi(x)$ denotes the standard normal density function;  see
-[@Feller1968], [@NavarroFuss2009].
+\beta) \phi \! \left(\frac{(2k + \beta)\alpha }{\sqrt{y - \tau}}\right),
+\end{equation*} where $\phi(x)$ denotes the standard normal density function;  
+
+and one representation that converges fast for large reaction-time values ("large time expansion"):
+\begin{equation*}
+\text{Wiener}(y|\alpha, \tau, \beta, \delta) =
+\frac{\pi}{\alpha^2} \exp \! \left(- \delta \alpha \beta -
+\frac{\delta^2(y-\tau)}{2}\right) \sum_{k = 1}^{\infty} k \exp \! \left(-\frac{k^2\pi^2(y-\tau)};
+{2\alpha^2}\right) \sin \!(k\pi\beta)
+\end{equation*}
+see [@Feller1968], [@NavarroFuss2009].
 
 ### Distribution statement