On the strong law of large numbers for ϕ-sub-Gaussian random variables

  • K. Zajkowski Inst. Math., Univ. Bialystok, Poland
Keywords: ϕ-subgaussian random variables, strong law of large numbers


UDC 517.9

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 $\inf\{a\ge 0\colon \forall_{\lambda\in\mathbb{R}}$  $\ln\mathbb{E}\exp(\lambda\xi)\le\varphi_p(a\lambda)\};$ $\tau_{\varphi_p}$ is a norm in a space ${\rm Sub}_{\varphi_p}=\{\xi\colon \tau_{\varphi_p}(\xi)<\infty\}$ of $\varphi_p$-sub-Gaussian random variables. We prove that if for a sequence $(\xi_n)\subset {\rm Sub}_{\varphi_p},$  $p>1,$ there exist positive constants $c$ and $\alpha$ such that for every natural number $n$  the inequality $\tau_{\varphi_p} \Big(\sum_{i=1}^n\xi_i \Big)\le cn^{1-\alpha}$ 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 strong law of large numbers for independent sub-Gaussian random variables [see   R. L. Taylor, T.-C. Hu, Sub-Gaussian techniques in proving strong laws of large numbers, Amer. Math. Monthly, 94, 295–299 (1987)] to the case of dependent $\varphi_p$-sub-Gaussian random variables.


K. Azuma, Weighted sums of certain dependent random variables , Tokohu Math. J., 19 , 357 – 367 (1967), https://doi.org/10.2748/tmj/1178243286

A. Bulinski, A. Shashkin, Limit theorems for associated random fields and related systems , World Sci. Publ. (2007), https://doi.org/10.1142/9789812709417

V. Buldygin, Yu. Kozachenko, Metric Characterization of Random Variables and Random Processes , Amer.Math.Soc., Providence, RI, (2000), https://doi.org/10.1090/mmono/188

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

R. Giuliano Antonini, Yu. Kozaczenko, A. Volodin, Convergence of series of dependent φ -sub-Gaussian random variables , J. Math. Anal. Appl. 338, 1188 – 1203 (2008), https://doi.org/10.1016/j.jmaa.2007.05.073

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

W. Hoeffding, Probability for sums of bounded random variables , J. Amer. Statist. Assoc., 58 , 13 – 30 (1963).

J.P. Kahane, Local properties of functions in terms of random Fourier series (in French) , Stud. Math., 19 , № 1, 1 – 25 (1960), https://doi.org/10.4064/sm-19-1-1-25

R.L. Taylor, T.-C. Hu, Sub-Gaussian techniques in proving strong laws of large numbers , Amer. Math. Monthly, 94 , 295 – 299 (1987), https://doi.org/10.2307/2323401

K. Zajkowski, On norms in some class of exponential type Orlicz spaces of random variables , arXiv:1709.02970v2.

How to Cite
ZajkowskiK. “On the Strong Law of Large Numbers for ϕ-Sub-Gaussian Random Variables”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 3, Mar. 2021, pp. 431 -36, doi:10.37863/umzh.v73i3.197.
Short communications