0

In my PhD thesis I need to discuss both standard deviation and sigma algebra. I’m currently using $\hat X$ to denote the sigma algebra, because expressions such as the $\hat X \cap \hat Y$ look pretty, but I’m inventing notation perhaps unnecessarily.

Should I avoid using sigma to denote both these? Also, how might one interpret $\sigma(S)$ — is it a sigma algebra or the standard deviation?

Advice welcome.

gen-ℤ ready to perish
  • 6,732
Jim Newton
  • 519
  • 1
    Maybe use $\sigma_X$ to denote standard deviation and $\sigma(X)$ for the sigma algebra generated by $X$. – Calculon Apr 18 '18 at 10:45
  • I'm currently using $\widehat{X}$ to denote the sigma algebra, because expressions such as the following look pretty, $\widehat{X} \cap \widehat{Y}$ but I'm inventing notation perhaps unnecessarily. – Jim Newton Apr 18 '18 at 11:02
  • Then I don't understand your question. If you already have a separate notation for sigma algebras distinctly different from the standard deviation symbol, what is the issue here? – Calculon Apr 18 '18 at 11:10
  • The issue is that I invented a notation to supersede an existing notation. PhD thesis reviewers can be nitpicky, for good reason. I was basically asking what other experts might suggest. If there's a better standard notation, I'm happy to consider it. – Jim Newton Apr 18 '18 at 11:22
  • 2
    There is no mention of this new notation in your question. – Calculon Apr 18 '18 at 11:23
  • There's no mention of this new notation on my question because Stackexchange does not allow people such as myself to edit his questions, therefore I explained the alternate notation in a comment. Sorry if that's confusing. – Jim Newton Apr 18 '18 at 11:30
  • @JimNewton If you are logged into the same account you used to post this question (and it seems like you are), then you should be able to edit your question to include your proposed notation for $\sigma$-algebras. – Theoretical Economist Apr 18 '18 at 12:11
  • @Calculon I resolved the issue :) – gen-ℤ ready to perish Apr 18 '18 at 23:02

1 Answers1

1

I see no reason why you cannot use $s$ for standard deviation. I have seen this plenty, and the first site that came up when I Googled “standard deviation notation” suggests the notation $s$ as well. This is what I would recommend.

Alternatively (as is also mentioned on the site), you could use $\sigma_S$ for standard deviation and $\sigma(S)$ for sigma algebra. I don’t suppose you would want to resort to $\sqrt{\operatorname{Var}(S)}$ for standard deviation though.

At the end of the day, you are a PhD student, so you understand thoroughly how notation is totally arbitrary, but I understand your desire to make your thesis as unambiguous as possible. Keep in mind that your readership will be comprised of intelligent academics, just like yourself. I think they’ll be able to handle whatever notation you define, especially when it appears in context and when you aren’t afraid to use your words (i.e., don’t rely entirely on notation).

gen-ℤ ready to perish
  • 6,732