Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities
Abstract
The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> moment of a sum of independent symmetric random variables to that of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> moments of the individual variables, are computed in the range <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 greater-than p less-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2>p\le 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This complements the work of Utev who has done the same for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 4"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p>4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The qualitative nature of the extreme cases turns out to be different for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 4"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p>4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> than for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 4"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p>4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The method developed yields results in some more general and other related moment inequalities.