Asymptotic properties of self-normalized linear processes with long memory
The authors study the convergence to fractional Brownian motion for long memory time series having independent innovations with infinite second moments. Their study is motivated by economic applications, e.g., the so-called unit root testing problem. Thus the process investigated is NEWLINE\[NEWLINE X_k = \sum_{i=0}^{\infty} a_i \epsilon_{k-i}, \quad k \geq 1, NEWLINE\]NEWLINE where the random variables \(\{\epsilon_n : n \in \mathbb{Z}\}\) are i.i.d., centered and in the domain of attraction of a normal law, that is, \(\ell(x):=\int_{|\epsilon_n| \leq x} \epsilon_n^2 d\mathbb{P}\) is a slowly varying function at \(\infty\). It is assumed that the real coefficients \(\{a_n : n \geq 0\}\) decrease as \(a_n = n^{-\alpha} L(n)\), where \(1/2 < \alpha < 1\), and \(L\) is a slowly varying function at \(\infty\).NEWLINENEWLINEThe main result of the paper is that the process \(W_n(t) := S_{\lfloor nt\rfloor} /B_n\), \((0 \leq t \leq 1)\) weakly converges to the fractional Brownian motion \(W_H\) with Hurst index \(H=3/2 -\alpha\), on the Skorokhod space \(D[0,1]\), where \(S_n = \sum_{i=1}^n X_i\) and \(B_n\) is a normalizing constant defined in the paper. Since \(B_n\) depends on the function \(\ell\) which is not known in general, the authors give a self-normalized version of the theorem: NEWLINE\[NEWLINE \frac{S_{\lfloor nt\rfloor}}{n a_n \left(\sum_{i=1}^n X_i^2\right)^{1/2}} \Rightarrow \frac{\sqrt{c_{\alpha}}}{A} W_H(t) , NEWLINE\]NEWLINE where the constants \(c_\alpha\) and \(A\) are defined in the paper and \(\Rightarrow\) denotes weak convergence.
scientific article; zbMATH DE number 6045071
scientific article