## On the strong law of large numbers for $\varphi$-subgaussian random variables

Preprint English OPEN
Zajkowski, Krzysztof (2016)
• Subject: 60F15 | Mathematics - Probability

For $p\ge 1$ let $\varphi_p(x)=x^2/2$ if $|x|\le 1$ and $\varphi_p(x)=1/p|x|^p-1/p+1/2$ if $|x|>1$. For a random variable $\xi$ let $\tau_{\varphi_p}(\xi)$ denote $\inf\{a\ge 0:\;\forall_{\lambda\in\mathbb{R}}\; \ln\mathbb{E}\exp(\lambda\xi)\le\varphi_p(a\lambda)\}$; $\tau_{\varphi_p}$ is a norm in a space $Sub_{\varphi_p}=\{\xi:\;\tau_{\varphi_p}(\xi)<\infty\}$ of $\varphi_p$-subgaussian random variables. We prove that if for a sequence $(\xi_n)\subset Sub_{\varphi_p}$ ($p>1$) there exist positive constants $c$ and $\alpha$ such that for every natural number $n$ the following inequality $\tau_{\varphi_p}(\sum_{i=1}^n\xi_i)\le cn^{1-\alpha}$ holds then $n^{-1}\sum_{i=1}^n\xi_i$ converges almost surely to zero as $n\to\infty$. This result is a generalization of the SLLN for independent subgaussian random variables (Taylor and Hu \cite{TayHu}) to the case of dependent $\varphi_p$-subgaussian random variables.
• References (7)

 A. Bulinski, A. Shashkin, Limit Theorems for Associated Random Fields and Related Systems, World Scientific Publishing Co. Pte. Ltd. 2007.

 V. Buldygin, Yu. Kozachenko, Metric Characterization of Random Variables and Random Processes, Amer.Math.Soc., Providence, RI, 2000.

 V. Buldygin, Yu. Kozachenko, Subgaussian random variables, Ukrainian Math. J. 32 (1980), 483-489.

 R. Giuliano Antonini, Yu. Kozaczenko, A. Volodin, Convergence of series of dependent ϕ-subgaussian random variables, J. Math. Anal. Appl. 338(2008) 1188-1203.

 J.-B. Hiriart-Urruty, C. Lemar´echal, Convex Analysis and Minimization Algorithms. II, Springer-Verlag, Berlin Heidelberg 1993.

 J.P. Kahane, Local properties of functions in terms of random Fourier series (French), Stud. Math., 19 (no. 1), 1-25 (1960).

 R.L. Taylor, T.-C. Hu, Sub-Gaussian techniques in proving strong laws of large numbers, Amer. Math. Monthly 94 (1987) 295-299.

• Similar Research Results (9)
 Bifurcation from infinity and nodal solutions of quasilinear elliptic differential equations (2014) 95% On the Azuma inequality in spaces of subgaussian of rank $p$ random variables (2017) 93% Electrically assisted light-induced gliding of nematic liquid-crystal easy axis at varying polarization azimuth of reorienting light: model and experiment (2012) 93% Exact number of solutions for a Neumann problem involving the p-Laplacian (2014) 88% Semiclassical Resonances of Schrödinger operators as zeroes of regularized determinants (2008) 86% Convergence of functionals and its applications to parabolic equations (2004) 77% Infinitely many homoclinic solutions for the second-order discrete $p$-Laplacian systems (2013) 73% EXISTENCE OF HOMOCLINIC SOLUTIONS FOR THE SECOND-ORDER DISCRETE P-LAPLACIAN SYSTEMS (2011) 73% Multiplicity of solutions for quasilinear elliptic boundary-value problems (1999) 70%
• Metrics
No metrics available
Share - Bookmark