Joint convergence along different subsequences of the signed cubic variation of fractional Brownian motion II

David Nualart,Jason Swanson

doi:10.1214/ecp.v18-2840

Abstract

The purpose of this paper is to provide a complete description the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brownian motion with Hurst parameter $ H = 1 / 6$.

Highlights

Suppose that B = {B(t), t ≥ 0} is a fractional Brownian motion with Hurst parameter H=Let ⌊x⌋ denote the greatest integer less than or equal to x.In [6], Nualart andOrtiz-Latorre proved that the sequence of sums, ⌊nt⌋Wn(t) = (B(j/n) − B((j − 1)/n))3, j=1 converges in law to a Brownian motion W = {W (t), t ≥ 0}, with variance κ2t given by κ2 = 3 (|m + 1|1/3 + |m − 1|1/3 − 2|m|1/3)3. 4 m∈ZThe process W is related to the signed cubic variation of B
In [1], Burdzy, Nualart and Swanson studied the convergence in distribution of the sequence of two-dimensional processes {(Wan(t), Wbn(t))}, where {an}∞ n=1 and {bn}∞ n=1 are two strictly increasing sequences of natural numbers converging to infinity
The purpose of this paper is to provide a complete description of the asymptotic behavior of Wan(t) and Wbn(t) for all sequences {an} and {bn}

Summary

Introduction

(B, Wan, Wbn) ⇒ (B, Xρ) in the Skorohod space DR3[0, ∞) as n → ∞, in the following cases: (i) The set Ic is finite (which implies L ∈ Q). Theorem 1.1 covers many simple and interesting pairs of sequences, and helps to tell a surprising story about the asymptotic correlation between the sequences, {Wan} and {Wbn}, both of which are converging to a Brownian motion. Many sequences whose ratios converge to an irrational number are not covered by this theorem. Let L = p/q, where p, q ∈ N are relatively prime numbers.

Objectives

Results

Conclusion