Should random vectors be Capital? I'm reading lot's of article in high-dimensional statistics and in most of them random vectors written in small characters, while I thought always random vectors written in Capital. Are they using wrong notation or it's a normal thing?
Any help will be appreciated.
 A: Capitalisation of random variables/vectors is a matter of convention, but it is important to understand that the convention has a purpose.  Probability and statistics frequently deals with cases where a quantity is treated as random in some contexts (e.g., when it has not yet been observed) but fixed in other contexts (e.g., when it has been observed, or we are conditioning on it for some reason).  The purpose of capitalisation of random vectors is to ensure that the reader is able to quickly differentiate between a quantity that is being treated as random, and that same quantity once it is treated as a fixed value.  With this in mind, you will find that the notational conventions differ in different types of discussions of probability and statistics.
Classical statistics: In this methodology, quantities can be either random or fixed depending on the context.  Hence, it is common to observe the capitalisation convention in order to make it clear when a quantity is being treated as random and when it is being treated as fixed.
Bayesian statistics: In this methodology, any quantity that is unknown/unobserved is treated as random and any quantity that is known/observed is treated as fixed.  All quantities that are capable of being treated as random variables enter into probability statements either in the main argument or in the conditioning argument, and so the distinction between random variables and constants is usually clear from the probability statements, even without any capitalisation.  For this reason, Bayesian treatments often eschew the capitalisation convention and use only lower-case letters.
Statistics with vectors and matrices: When statistical discussions use matrices there is an additional complication because it is also a common notational convention to represent matrices with upper-case letters and vectors with lower-case letters.  Here the two notational conventions collide, and different authors deal with this in different ways.  If the probabilistic content of the discussion is the most important, usually the capitalisation convention is observed, but if the linear algebra content is most important then sometimes the convention is eschewed in favour of the capitalisation convention for linear algebra.
