Weak laws of large numbers for maximal weighted sums of random variables

In this paper, we establish a weak law of large numbers for a class of weighted sums of random variables introduced by Jajte (2003 Jajte, R. 2003. On the strong law of large numbers. The Annals of Probability 31 (1):409–12. URL: http://www.jstor.org/stable/3481585. doi: 10.1214/aop/1046294315.[Crossref], [Web of Science ®] , [Google Scholar]). The obtained method allows us to deduce a generalized version of the Marcinkiewicz-Zygmund weak law of large numbers and to strengthen several known results, such as those of Gut (2004 Gut, A. 2004. An extension of the Kolmogorov-Feller weak law of large numbers with an application to the St. Petersburg game. Journal of Theoretical Probability 17 (3):769–79. doi: 10.1023/B:JOTP.0000040299.15416.[Crossref], [Web of Science ®] , [Google Scholar]) and Naderi et al. (2018 Naderi, H., P. Matuła, M. Amini, and H. Ahmadzade. 2018. A version of the Kolmogrov-Feller weak law of large numbers for maximal weighted sums of random variables. Communications in Statistics-Theory and Methods :1. doi: 10.1080/03610926.2018.1513146.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Finally, an application to randomly indexed sums is presented.