where w is defined as in (59). We are now ready to compute the limits of the upper bound (49) and lower bound (50). It turns out that the limits of these bounds coincide and we receive Note that the special case z = − 2 corresponds to p = q and gives the coefficient − 4 / 3 of x $^{4}$. The calculations were performed with Mathematica and are much to unwieldy to fit in this paper. We have prepared a Mathematica notebook that performs the calculations step by step, using some practical transformation rules. What we compute is actually the limit of the logarithm of the upper and lower bound. The steps are as follows; compute series expansions of the different powers of r = q/p , use them inside the dilogarithms and then compute their series expansions, add the dilogarithms and the series expansions of the logarithms of the remaining factors. Some transformations of this expression helps Mathematica to take the limit that gives the result. Figure 1 demonstrates how the asymptotic ratio is achieved with increasing n . It shows R$_{p,q}$ ( n, n/ 2 , x n 3 / $^{4}$) for z = − 9 at y = 2 w = − 0 . 10539 . . . and the asymptotic ratio is given by (62), that is, e − 0 . 719718 x $^{4}$. The red curve is the asymptote and the blue curves are for finite n where the curves for larger n are closer to the asymptote. Changing p only slightly, say on the order of 1 /n 3 / $^{2}$, allows us to move around in the region where the distribution is wide. The end result is that with q = 1 + z/n and p = 1 + 2 w/n + a/n 3 / 2 we get The ratio expression gets nicer if we change the coefficient of 1 /n 3 / $^{2}$. We suggest the following parametrization instead; with q = 1 + z/n let and we receive R$_{p,q}$ ( n, n/ 2 , x n 3 / $^{4}$) ∼ exp ( w z (2 w + z ) x 4 6 ) (62) R$_{p,q}$ ( n, n/ 2 , x n 3 / $^{4}$) ∼ exp ( w z (2 w + z ) x 4 6 + a z (1 + w ) x 2 2 w − z ) (63) p = 1 + 2 w n + a w ( z 2 − 4 w $^{2}$) 3 (1 + w ) n 3 / 2 (64) R$_{p,q}$ ( n, n/ 2 , x n 3 / $^{4}$) ∼ exp ( w z (2 w + z ) 6 ( x 4 − 2 a x $^{2}$) ) (65) 19 B. Wide and double-peaked distributions