Sharp ratio inequalities for a conditionally symmetric martingale

• Autorzy: Osękowski A.
• Języki publikacji: EN

Abstrakty

EN

Let ƒ be a conditionally symmetric martingale and let S(ƒ) denote its square function. (i) For p, q > 0, we determine the best constants Cp,q such that [wzór...]. Furthermore, the inequality extends to the case of Hilbert space valued ƒ. (ii) For N = 1,2,... and q > 0, we determine the best constants C'N,q such that [wzór...]. These bounds are extended to sums of conditionally symmetric variables which are not necessarily integrable. In addition, we show that neither of the inequalities above holds if the conditional symmetry is not assumed.

Słowa kluczowe

EN

martingale; square function; ratio inequality; self-normalized process

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2010
• Tom: Vol. 58, no 1
• Strony: 65--77
• Opis fizyczny: Bibliogr. 18 poz.

Twórcy

autor

Osękowski A.

• Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2 02-097 Warszawa, Poland,

Bibliografia

• [Ab] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Reprint of the 1972 edition, Dover Publ., New York, 1992.
• [BT] B. Bercu and A. Touai, Exponential inequalities for self-normalized martingales with applications, Ann. Appl. Probab. 18 (2008), 1848-1869.
• [Bu] D. L. Burkholder, Explorations in martingale theory and its applications, in: École d’Été de Probabilités de Saint-Flour XIX-1989, Lecture Notes in Math. 1464, Springer, Berlin, 1991, 1-66.
• [D] B. Davis, On the Lp norms of stochastic integrals and other martingales, Duke Math. J. 43 (1976), 697-704.
• [D1] V. H. De La Pena, A general class of exponential inequalities for martingales and ratios, Ann. Probab. 27 (1999), 537-564.
• [D2] V. H. De La Pena, M. J. Klass and T. L. Lai, Self-normalized processes: Exponential inequalities, moment bounds and iterated logarithm law, Ann. Probab 32 (2004), 1902-1933.
• [DM] C. Dellacherie and P. A. Meyer, Probabilities and Potential B, North-Holland, Amsterdam, 1982.
• [K] A. Khintchine [A. Ya. Khinchin], Über dyadische Brüche, Math. Z. 18 (1923), 109-116.
• [L] J. E. Littlewood, On bounded bilinear forms in an infinite number of variables, Quart. J. Math. Oxford 1 (1930), 164-174.
• [M] J. Marcinkiewicz, Quelques théorèmes sur les séries orthogonales, Ann. Soc. Polon. Math. 16 (1937), 84-96.
• [MZ] J. Marcinkiewicz et A. Zygmund, Quelques théorèmes sur les fonctions indépendantes, Studia Math. 7 (1938), 104-120.
• [O] A. Osękowski, Two inequalities for the first moment of a martingale, its square and maximal function, Bull. Polish Acad. Sci. Math. 53 (2005), 441-449.
• [P] R. E. A. C. Paley, A remarkable series of orthogonal functions I, Proc. London Math. Soc. 34 (1932), 241-264.
• [PP] J. L. Pedersen and G. Peskir, Solving non-linear optimal stopping problems by the method of time-change, Stochastic Anal. Appl. 18 (2000), 811-835.
• [PX] G. Pisier and Q. Xu, Non-commutative martingale inequalities, Comm. Math. Phys. 189 (1997), 667-698.
• [S1] E. M. Stein, The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. 7 (1982), 359-376.
• [S2] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.
• [W] G. Wang, Sharp square-function inequalities for conditionally symmetric martingales, Trans. Amer. Math. Soc. 328 (1991), 393-419.