A maximal theorem of Hardy-Littlewood type for pairwise i.i.d. and the law of large numbers

Пусть $p\in [1,2)$. Мы покажем, что если $(X_n)_{n=1}^\infty$ - последовательность попарно независимых одинаково распределенных случайных величин с $\mathbf{E}|X_1|^p<\infty$, то $$ \mathbf{P}[\sup_n|\frac{S_n}{n^{1/p}}|> \alpha]\le \frac{C_p \mathbf{E}|X_1|^p}{\alpha^p}\quadдля любого \alpha>0, $$ где $C_p$ - некоторая константа, зависящая от $p$, и $S_n:=\sum_{i=1}^n(X_i-\mathbf{E}X_i)$. При доказательстве мы воспользовались следствием более общего утверждения, в котором требовалось только, чтобы последовательность $(X_n)$ была слабо коррелирована в смысле Рио. Мы докажем неравенство, дающее скорость сходимости $\lim_{n\to\infty}|S_n|/{n^{{1}/{p}}}=0$ п.н., и таким образом усилим основной результат [E. Rio, "Vitesses de convergence dans la loi forte pour des suites dépendantes", C. R. Acad. Sci. Paris Sér. I Math. 320:4 (1995), 469-474].

Some Properties of Quasinearly Subharmonic Functions and Maximal Theorem for Bergman Type Spaces

Let denote the class of quasinearly subharmonic functions in unit ball . We provide, following result: if and if , then , where is the radial maximal function and , and . Also, we prove a maximal theorem for Bergman type spaces.

A New Maximal Theorem in Product GFC-Spaces with Application to Systems of Generalized Mixed Vector Quasiequilibrium Problems

In this paper, a new maximal element theorem is established in product GFC-spaces. As application, a new existence theorem of solutions for systems of generalized mixed vector quasi-equilibrium problems is obtained.

VECTOR-VALUED THEOREM FOR THE UNCENTRED MAXIMAL OPERATOR ON BESSEL–KINGMAN HYPERGROUPS

In this paper we introduce a vector-valued uncentred maximal operator in the setting of one-dimensional Bessel–Kingman hypergoups, and prove a maximal theorem for it.

On the relation of Carleson's embedding and the maximal theorem in the context of Banach space geometry

Hytönen, McIntosh and Portal (J. Funct. Anal., 2008) proved two vector-valued generalizations of the classical Carleson embedding theorem, both of them requiring the boundedness of a new vector-valued maximal operator, and the other one also the type $p$ property of the underlying Banach space as an assumption. We show that these conditions are also necessary for the respective embedding theorems, thereby obtaining new equivalences between analytic and geometric properties of Banach spaces.