A note on almost sure uniform and complete convergences of a sequence of random variables

作者：Cruz Rambaud Salvador

来源：Stochastics-An International Journal of Probability and Stochastic Reports, 2011, 83(3): 215-221.

DOI：10.1080/17442508.2010.526211

• convergence
• uniform
• almost sure
• in mean
• complete
• local boundedness

The aim of this note is to introduce another way of defining the almost sure uniform convergence, which is necessary when studying some mathematical results on the existence of price bubbles in certain scenarios of trading securities. This mode of convergence of random variables' sequences is intermediate between the uniform and the almost sure ones, and, more specifically, between the uniform and the complete convergences. In this way, this paper presents some mathematical characterizations of both almost sure uniform and complete convergences, and shows that the almost sure uniform convergence is a particular case of complete convergence, when the number of summands in the series defining this mode of convergence is finite. Finally, this paper presents the relation of almost surely uniform convergence with convergence in mean when the random variable limit is integrable. Moreover, almost surely convergence and local boundedness of the sequence of random variables minus its limit are sufficient to derive convergence in mean.

• 出版日期2011