by Jotun Hein, Mikkel H. Schierup, and Carsten Wiuf
ISBN 0-19-852996-1
The \((1 - 1/(2 N))\) term should be omitted
The \(\approx\) should be replaced with \(=\).
The third sigma should be \[ \sum_{j=2}^i \]
The equation should be \[ E(R_n^3) \approx \left( \log(n) - 1.56 \right) \rho \]
From Professor Carsten Wiuf:
In the two sex model, one forms pairs of males and females, so going backwards, two genes come from the female population with probability \(1/4\) and the male with probability \(1/4\). If from female population, the two genes have probability \(1/(2N_f)\) for common ancestor and similar when males. So this yields
\[ \frac{1}{4} \left( \frac{1}{2N_f} + \frac{1}{2N_m} \right) = \frac{1}{8} \frac{N}{N_f N_m} \]
From A Simulation Study of the Reliability of Recombination Detection Methods: \[ \frac{ E(R_n^3) }{\rho} = \sum_{i=1}^{n-1} \frac{1}{i} - C(n) \] and \[ C(n) \approx 2.14 \]
From the well known Euler-Mascheroni constant \(\gamma\): \[ \sum_{i=1}^n \frac{1}{i} \approx \log(n) + \gamma \approx \log(n) + 0.577 \]
Thus \[ E(R_n^3) \approx \left( \log(n) + 0.577 - 2.14 \right) \rho \]