Functional central limit theorems for seeds in a linear birth and growth model

A problem of heredity of mixing properties ($\alpha $-mixing, $\beta $-mixing and $\rho $-mixing) from a stationary point process on $\mathbb R\times \mathbb R_{+}$ to a sequence of some of its points called ‘seeds’ is considered. Next, using the mixing properties, several versions of functional central limit theorems for the distances between seeds and the process of the number of seeds are obtained.