Understanding Limit Theorems for Semimartingales: A Short Survey

Abstract

This paper presents a short survey on limit theorems for certain functionals of semimartingales, which are observed at high frequency. Our aim is to explain the main ideas of the theory to a broader audience. We introduce the concept of stable convergence, which is crucial for our purpose. We show some laws of large numbers (for the continuous and the discontinuous case) that are the most interesting from a practical point of view, and demonstrate the associated stable central limit theorems. Moreover, we state a simple sketch of the proofs and give some examples.

A central limit theorem for the realised covariation of a bivariate Brownian semistationary process

“…Before stating our central limit theorem result, we need to briefly introduce the notion of stable convergence, which is the type of convergence that we will encounter, and which is typically used in inference for semimartingales. In this section we take definitions and results from Aldous and Eagleson (1978) and from the survey on uses and properties of stable convergence in Podolskij and Vetter (2010).…”

The null hypothesis of (common) jumps in case of irregular and asynchronous observations

“…, 4 are standard normally distributed random variables which are independent of F and of the random vectors (L 1 , R 1 , L 2 , R 2 , L, R)(s). (6.41) together with Proposition 2.2 in Podolskij and Vetter (2010)…”

“…p,− , ξ k,+ (S q,p )U Sq,p,+ Sq,p≤T for standard normally distributed random variables (U s,− , U s,+ ) which are independent of F and of the ξ k,− (s), ξ k,+ (s). Using this stable convergence, Proposition 2.2 inPodolskij and Vetter (2010) and the continuous mapping theorem we then obtain 4 (σ(r, q) Sq,p− ∆ in(Sq,p)−j,n W +σ(r, q)S q,p ∆ in(Sq,p)+j,n W )…”

EXcess Idle Time

“…We note that the application of Lemma 3 hinges on the following derivation: We now turn to the limiting distribution. We will verify the conditions in Theorem 3-1 (page 243) of Jacod (1997), as reviewed in Podolskij and Vetter (2010), Theorem 1. In what follows, we will therefore refer to the numbering of the conditions in Theorem 1 of Podolskij and Vetter (2010), henceforth PV (2010).…”