Lai law for linear processes with long memory (Q6618764)

At the very beginning the authors recall the following result of Lai: Let\N\(r > 1\)\Nand\N\(\left\{ {{X_n},n \ge 1} \right\}\)\Nbe a sequence of i.i.d random variables with\N\(E{X_1} = 0\)\Nand\N\(EX_1^2 = 1.\)\NLet also\N\({S_n} = \sum_{k = 1}^n {{X_k}},\)\N\(n \ge 1.\)\N\NSuppose that\N\(E{\left( {X_1^2/\log \left| {{X_1}} \right|} \right)^r} < \infty,\)\Nwhere\N\(\log x = \ln \max \{ x;e\}.\)\NThen for all\N\(\varepsilon > \sqrt {r - 1},\)\N\[\N\sum\limits_{n = 1}^\infty {{n^{r - 2}}P\left\{ {\left| {{S_n}} \right| > \varepsilon \sqrt {2n\log n} } \right\}} < \infty. \tag{*}\N\]\NConversely, if (*) holds for some\N\(\varepsilon > 0,\)\Nthen\N\(E{X_1} = 0\)\Nand\N\(E{\left( {X_1^2/\log \left| {{X_1}} \right|} \right)^r} < \infty.\)\N\NThe authors generalize this result to the linear processes with long memory. Let be a sequence of i.i.d. random variables and a sequence of real numbers. The sequence\N\(\{ {X_n},n \ge 1\}\)\Nis called a linear process, if\N\[\N{X_n} = \sum\limits_{i = - \infty }^\infty {{a_{i + n}}{\zeta _i}}.\N\]\NIf\N\(\sum_{i = - \infty }^\infty {\left| {{a_i}} \right|} < \infty,\)\N\(\sum_{i = - \infty }^\infty {{a_i}} \ne 0,\)\N\(\{ {X_n},n \ge 1\}\)\Nis a linear process with short memory. If\N\(\sum_{i = - \infty }^\infty {\left| {{a_i}} \right|} = \infty,\)\N\(\{ {X_n},n \ge 1\}\)\Nis a linear process with long memory. The article considers the latter case.