| Summary: | In this paper, we consider the mildly explosive autoregression <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>y</mi><mi>t</mi></msub><mo>=</mo><msub><mi>ρ</mi><mi>n</mi></msub><msub><mi>y</mi><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>u</mi><mi>t</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mi>n</mi></mrow></semantics></math></inline-formula>, where <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><mo>+</mo><mi>c</mi><mo>/</mo><msup><mi>n</mi><mi>ν</mi></msup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>u</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula> are arithmetically <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-mixing errors. Under some weak conditions, such as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>E</mi><mo>|</mo></mrow><msub><mi>u</mi><mn>1</mn></msub><msup><mrow><mo>|</mo></mrow><mrow><mn>4</mn><mo>+</mo><mi>δ</mi></mrow></msup><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula> for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and mixing coefficients <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><msup><mi>n</mi><mrow><mo>−</mo><mo>(</mo><mn>2</mn><mo>+</mo><mn>8</mn><mo>/</mo><mi>δ</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the Cauchy limiting distribution is established for the least squares (LS) estimator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mi>n</mi></msub></semantics></math></inline-formula> 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>, which extends the cases of independent errors and geometrically <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-mixing errors. Some simulations for <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>, such as the empirical probability of the confidence interval and the empirical density, are presented to illustrate the Cauchy limiting distribution, which have good finite sample performances. In addition, we use the Cauchy limiting distribution of the LS estimator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi>ρ</mi><mo>^</mo></mover><mi>n</mi></msub></semantics></math></inline-formula> to illustrate real data from the NASDAQ composite index from April 2011 to April 2021.
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