Role of Counterexamples in Probability, Statistics and their Applications

Abstract

For objects such as random events, variables, vectors, processes, samples, etc., under specific assumptions, we prove specific results. It is more than important, also curious,  to know the range of validity, hence the range of applicability, of any available result. This is why we pay much attention to the counterexamples, always telling us that something is wrong. They provide answers to tricky questions on the role of the conditions under which a result is true or is not true, the (in)validity of converse statements, etc. 

I will suggest effective illustrations of independence, characteristics of distributions, limit theorems (LLN, CLT, LIL), moments, characterizations, statistical inference. 

In addition to what everybody knows, several unexpected/surprising facts will be presented.  The goal of the counterexamples is to enhance our knowledge, prevent from making wrong steps, helping to keep our eyes open. My appeal is: Pay attention to what is available, ask questions, on any new occasion, try to answer, discuss with others. This will lead to better theory and its correct applications. Also, these will bring to you intellectual pleasure.