Weak Convergence of Sums of Moving Averages in the $\alpha$-Stable Domain of Attraction

Heavy-Tailed Distributions and Robustness in Economics and Finance

“…As in McElroy and Politis (2002), we provide the results for the AR(1) model with the autoregressive parameter 0.5 and an MA(11) process for the sample sizes T D 100; 1000 and the tail index equal to 1.2 and 1.5. 16 It is important to note that, despite this convergence of finite-dimensional distributions, the process d 1 T P OETr tD1 .X t / does not converge weakly to S .r/ in DOE0; 1 endowed with the usual Skorohod topology (see Avram andTaqqu 1992 andRemarks 3.20 in Phillips andSolo 1992). 17 The fact that, in this approach, the inference is based on the t-statistic for block sample means is in contrast to subsampling that uses the empirical distribution function of t-statistics calculated over the block of size b as an approximation to the limit distribution of the full-sample t-statistic t T D p…”

A Lindeberg–Feller theorem for stable laws

“…These approximations are useful for the dual purposes of intuition and simulation of stable processes. We note that convergence in fdd is hard to improve upon since Avram and Taqqu (1992) showed that tightness of discretized moving averages (an example of a process given by M α [f t ]) cannot be achieved in the J 1 -Skorokhod topology.…”

On functional limits of short- and long-memory linear processes with GARCH(1,1) noises

“…noises have been extensively studied. See, for instance, [1,20,21,32,3,29,27,48,6,43]. To the best of our knowledge, the FLD for heavy-tailed linear processes with GARCH noise are new.…”