Geodesic
Maximal $l\phi$-inequalities for nonnegative submartingales
Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 1, pp. 162-172 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $(M_n)_{n\ge 0}$ be a nonnegative submartingale and let $M_n^*\stackrel{\textrm{def}}{=}\max_{0\le k\le n}M_k$, $n\ge 0$, be the associated maximal sequence. For nondecreasing convex functions $\phi\colon[0,\infty)\to[0,\infty)$ with $\phi(0)=0$ (Orlicz functions) we provide various inequalities for $E\phi(M_n^*)$ in terms of $E\Phi_a(M_n)$, where, for $a\ge 0$, $$ \Phi_{a}(x)\,\stackrel{\textrm{def}}{=}\,\int_{a}^{x}\!\!\int_{a}^{s}\frac{\phi'(r)}{r}\,dr\,ds, \qquad x>0. $$ Of particular interest is the case $\phi(x)=x$ for which a variational argument leads us to $$ EM_n^*\le\Bigg(1+\bigg(E\bigg(\int_{1}^{M_n\vee 1}\log x\,dx\bigg)\bigg)^{1/2}\Bigg)^2. $$ A further discussion shows that the given bound is better than Doob's classical bound $e(e-1)^{-1}(1+\textbf E M_n\log^{+}M_n)$ whenever $\textbf E(M_n-1)^{+}\ge e-2\approx 0.718$.
Keywords: nonnegative submartingale, Orlicz function, Young function, convex function inequality.
Mots-clés : maximal sequence, Choquet representation 
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U. Rösler; G. Alsmeyer. Maximal $l\phi$-inequalities for nonnegative submartingales. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 1, pp. 162-172. http://geodesic.mathdoc.fr/item/TVP_2005_50_1_a10/

[1] Alsmeyer G., Rösler U., “The best constant in the Topchii–Vatutin inequality for martingales”, Statist. Probab. Lett., 65:3 (2003), 199–206 | DOI | MR | Zbl

[2] Bingham N. H., Goldie C. M., Teugels J. L., Regular Variation, Cambridge Univ. Press, Cambridge, 1987, 491 pp. | MR | Zbl

[3] Burkholder D. L., “Explorations in martingale theory and its applications”, Lecture Notes in Math., 1464, 1991, 1–66 | MR | Zbl

[4] Chow Y. S., Teicher H., Probability Theory: Independence, Interchangeability, Martingales, Springer-Verlag, New York, 1997, 488 pp. | MR

[5] Dellacherie C., Meyer P. A., Probabilities and Potential. B. Theory of Martingales, North-Holland, Amsterdam, 1982, 463 pp. | MR | Zbl

[6] Gundy R. F., “On the class $L\log L$, martingales, and singular integrals”, Studia Math., 33 (1969), 109–118 | MR | Zbl

[7] Long R., Martingale Spaces and Inequalities, Peking Univ. Press, Beijing; Vieweg, Braunschweig, 1993, 346 pp. | MR | Zbl

[8] Neveu J., Discrete-Parameter Martingales, North-Holland, Amsterdam, 1975, 236 pp. | MR | Zbl