| Summary: | This paper considers stationary autoregressive (AR) models with heavy-tailed, general GARCH (G-GARCH) or augmented GARCH noises. Limit theory for the least squares estimator (LSE) of autoregression coefficient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ρ</mi><mo>=</mo><msub><mi>ρ</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> is derived uniformly over stationary values in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, focusing on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ρ</mi><mi>n</mi></msub><mo>→</mo><mn>1</mn></mrow></semantics></math></inline-formula> as sample size <i>n</i> tends to infinity. For tail index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula> of G-GARCH innovations, asymptotic distributions of the LSEs are established, which are involved with the stable distribution. The convergence rate of the LSE depends on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mi>n</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, but no condition on the rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mi>n</mi></msub></semantics></math></inline-formula> is required. It is shown that, for the tail index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>, the LSE is inconsistent, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><mi>n</mi><mo>/</mo><mo>(</mo><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mi>n</mi><mn>2</mn></msubsup><mo>)</mo></mrow></semantics></math></inline-formula>-consistent, and for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>n</mi><mrow><mn>1</mn><mo>−</mo><mn>2</mn><mo>/</mo><mi>α</mi></mrow></msup><mo>/</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msubsup><mi>ρ</mi><mi>n</mi><mn>2</mn></msubsup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>-consistent. Proofs are based on the point process and the asymptotic properties in AR models with G-GARCH errors. However, this present work provides a bridge between pure stationary and unit-root processes. This paper extends the existing uniform limit theory with three issues: the errors have conditional heteroscedastic variance; the errors are heavy-tailed with tail index <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></semantics></math></inline-formula>; and no restriction on the rate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>ρ</mi><mi>n</mi></msub></semantics></math></inline-formula> is necessary.
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