I am struggling with Theorem 3.5 from the paper of Davis and Mikosh "THE SAMPLE AUTOCORRELATIONS OF HEAVY-TAILED PROCESSES WITH APPLICATIONS TO ARCH". This paper is available, for example, here.
Context: authors use the notation $\varepsilon_\mathbf{x}$ for the Dirac measure at $\mathbf{x}$. Then they use the notation of $\varepsilon_{P_i}$ to denote the Poisson point process. In the screenshot below, $$\gamma_{n, X}(h) = n^{-1} \sum^{n-h}_{t=1} X_t X_{t+h}$$, where $X_t$ is a stochastic process.
I am struggling to understand the Theorem 3.5, which is presented below. I have completely no idea, what $P_i$ and $Q^{(h)}_{i,j}$ could mean in this setting in the definition of $V_h$.
The Corollary 2.4 is presented below.
Question: What could $P_i$ and $Q_{i,j}$ mean in this context (in the definition of $V_h$)? What is the intuition of such a representation, i.e. mean / variance?
Thanks a lot for any comments!