2021/04/14 by Alexander Iksanov, Iksanov, Alexander, Zakhar Kabluchko +3
Mathematics · Computer Science · Economics, Econometrics and Finance · #Stochastic processes and statistical mechanics #Bayesian Methods and Mixture Models #Financial Risk and Volatility Modeling
paper · pdf · doi:10.48550/arxiv.2104.06948
Let (pk)k\∈\ℕ be a discrete probability distribution for which\nthe counting function x\↦ # k\∈\ℕ: pk\≥ 1/x belongs to\nthe de Haan class \Π. Consider a deterministic weighted branching process\ngenerated by (pk)k\∈\ℕ. A nested Karlin's occupancy scheme is\nthe sequence of Karlin balls-in-boxes schemes in which boxes of the jth\nlevel, j=1,2,\… are identified with the jth generation individuals and\nthe hitting probabilities of boxes are identified with the corresponding\nweights. The collection of balls is the same for all generations, and each ball\nstarts at the root and moves along the tree of the deterministic weighted\nbranching process according to the following rule: transition from a mother box\nto a daughter box occurs with probability given by the ratio of the daughter\nand mother weights.\n Assuming there are n balls, denote by \Kn(j) the number of\noccupied (ever hit) boxes in the jth level. For each j\∈\ℕ, we\nprove a functional limit theorem for the vector-valued process\n(\K(1) lfloor eT+u rfloor,\…, \K(j) lfloor≠T+u rfloor)u\∈\ℝ, properly normalized and centered, as\nT\→\∞. The limit is a vector-valued process whose components are\nindependent stationary Gaussian processes. An integral representation of the\nlimit process is obtained.\n