Laws of large numbers for weighted sums of independent random variables: a game of mass

• Main Authors: Avena, Luca, da Costa, Conrado
• Format: Journal Article
• Language: English
• Published: 01-04-2020
• Subjects: Mathematics - Probability
• DOI: 10.48550/arxiv.2004.00469

We consider weighted sums of independent random variables regulated by an
increment sequence. We provide operative conditions that ensure strong law of
large numbers for such sums to hold in both the centered and non-centered case.
The existing criteria for the strong law are either implicit or assume some
sufficient decay for the sequence of coefficients. In our set up we allow for
arbitrary sequence of coefficients, possibly random, provided the random
variables regulated by such increments satisfy some mild concentration
conditions. In the non-centered case, convergence can be translated into the
behavior of a deterministic sequence and it becomes a game of mass provided the
expectation of the random variables is a function of the increments. We show
how different limiting scenarios can emerge by identifying several classes of
increments, for which concrete examples will be offered.