Wang, Zhichao (2018-05). Convergence of the Powers of Free Triangular Arrays to Higher Variations of Free L?vy Processes. Master's Thesis. Thesis uri icon

abstract

  • We focus on a free triangular array of random variables {XN,?}1<=?<=N , N ? N, in some noncommutative probability space (A, ?) such that the random variables are freely independent and identically distributed in each row. For each k ? N, our aim is to find some conditions to ensure the convergence of the sum of the k-th power of each row in the triangular array. We also want to know the expression of the limit random variable (the k-th variation), denoted by X(?) . The motivation of this study comes from the relation between the free stochastic measure of a free L?vy process and higher variations. First, when (A, ?) is a plain non-commutative probability space, we find out an equivalent condition for the joint convergence in distribution of all powers of the free triangular array. This condition requires the decay of all moments of the random variables in each row. Moreover, after defining the free stochastic measure of a free triangular array in terms of the convergence in distribution, we prove a free Kailath-Segall formula for centered stationary stochastic processes in our settings to describe the relationship between a stochastic measure and the higher variations. Second, when (A, ?) is a W*-probability space, considering all self-adjoint (possibly unbounded) operators affiliated with A, we prove that if the convergence in distribution of a free triangular array to a free L?vy process holds, then we have the convergence in distribution of all powers of the original triangular array towards all higher variations, which are also free L?vy processes. Moreover, the free L?vy-It? decomposition of each higher variation can be simplified by the L?vy-It? decomposition of the original L?vy process.
  • We focus on a free triangular array of random variables {XN,?}1<=?<=N , N ? N, in some noncommutative probability space (A, ?) such that the random variables are freely independent and identically distributed in each row. For each k ? N, our aim is to find some conditions to ensure the convergence of the sum of the k-th power of each row in the triangular array. We also want to know the expression of the limit random variable (the k-th variation), denoted by X(?) . The motivation of this study comes from the relation between the free stochastic measure of a free L?vy process and higher variations.

    First, when (A, ?) is a plain non-commutative probability space, we find out an equivalent condition for the joint convergence in distribution of all powers of the free triangular array. This condition requires the decay of all moments of the random variables in each row. Moreover, after defining the free stochastic measure of a free triangular array in terms of the convergence in distribution, we prove a free Kailath-Segall formula for centered stationary stochastic processes in our settings to describe the relationship between a stochastic measure and the higher variations.

    Second, when (A, ?) is a W*-probability space, considering all self-adjoint (possibly unbounded) operators affiliated with A, we prove that if the convergence in distribution of a free triangular array to a free L?vy process holds, then we have the convergence in distribution of all powers of the original triangular array towards all higher variations, which are also free L?vy processes. Moreover, the free L?vy-It? decomposition of each higher variation can be simplified by the L?vy-It? decomposition of the original L?vy process.

publication date

  • May 2018