Limit theorems for compact two-point homogeneous spaces of large dimensions

Abstract

Let$\mathbb{K}$ be the field ℝ, ℂ, or ℍ of real dimension ν. For each dimensiond≥2, we study isotropic random walks(Y ₁)₁≥0 on the projective space$\mathbb{P}^d (\mathbb{K})$ with natural metricD where the random walk starts at some$x_0^d \in \mathbb{P}^d (\mathbb{K})$ with jumps at each step of a size depending ond. Then the random variablesX ^d₁ :=cosD(Y ^d₁ ,x ^d₀ ) form a Markov chain on [−1, 1] whose transition probabilities are related to Jacobi convolutions on [−1, 1]. We prove that, ford→∞, the random variables (vd/2)(X ^d _l(d) +1) tend in distribution to a noncentralχ ²-distribution where the noncentrality parameter depends on relations between the numbers of steps and the jump sizes. We also derive another limit theorem for$\mathbb{P}^d (\mathbb{K})$ as well as thed-spheresS ^d ford→∞.