Second, the discrete dimensional-convolution procedure, which transforms the solution to the non-recursive expression of n, including a finite number of elementary operations and functions.
We derive a recursive expression for the expectation of the X-chromosomal admixture fraction as a function of sex-specific admixture parameters, demonstrating that the X-chromosomal admixture is obtained from a more complex formula than in the simple 2/3 and 1/3 weighting (Equation 1).
We derive an approximated recursive expression for the probability of the hop count with a given geographic distance.
Then, based on the connection between the number of perfect matchings and determinant of the skew adjacency matrix corresponding to a Pfaffian orientation, we derive the recursive expressions for the number of perfect matchings of the networks at two successive iterations.
Under a framework in which admixture occurs over time, potentially with different male and female contributions from the source populations, we have derived recursive expressions for the expectation, variance, and higher moments of the fraction of autosomal admixture.
The first step entails the factorization of our target distribution and manipulating it into a recursive expression (14).
We have not numerically analyzed complex time-varying admixture histories, but our recursive expressions flexibly accommodate a range of population histories, especially if simplifying assumptions are employed to reduce the number of parameters.
The closed-form and recursive expressions also agree with the results of the simulations (Table 3).
