Maqola

Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities

T. FigielInstitute of Mathematics, Polish Academy of Sciences, ul. Abrahama 18, 81â825 Sopot, PolandPaweł HitczenkoDepartment of Mathematics, North Carolina State University, Raleigh, North Carolina 27695â8205W. JohnsonDepartment of Mathematics, Texas A&M University, College Station, Texas 77843Gideon SchechtmanDepartment of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot, IsraelJoel ZinnDepartment of Mathematics, Texas A&M University, College Station, Texas 77843

1997en

ABI

Annotatsiya

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.

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DOI: 10.1090/s0002-9947-97-01789-3

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