- #1
fab13
- 300
- 7
- TL;DR
- I would like to know if we the expression of a double sum on the squared of a_lm (harmonical spherics from Legendre transformation) allows to include the sum of chi^2 factors in a chi^2 of sum (in the sense of "following distribution").
1) If I take as definition of ##a_{lm}## following a normal distribution with mean equal to zero and ##C_\ell=\langle a_{lm}^2 \rangle=\text{Var}(a_{lm})##, and if I have a sum of ##\chi^2##, can I write the 2 lines below (We use ##\stackrel{d}{=}## to denote equality in distribution).
Important remark : ##C_{\ell}## depends on ##\ell## : ##C_{\ell} = C_{\ell}(\ell)##
\begin{aligned}
\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2} & \stackrel{d}{=} \sum_{\ell=1}^{N}(2 \ell+1) \chi^{2}\left(C_{\ell}\right) \\
& \stackrel{d}{=} \chi^{2}\left(\sum_{\ell=1}^{N}(2 \ell+1) C_{\ell}\right)
\end{aligned}
that is to say, include the sum on ##N## into ##\chi^2## ?
Maybe I could write instead (with shape/scale convention for Gamma distribution):
##
\begin{aligned}
\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2} & \stackrel{d}{=} \sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} \Gamma\left(1 / 2,2 C_{\ell}\right) \\
& \stackrel{d}{=} \sum_{\ell=1}^{N} \Gamma\left((2 \ell+1) / 2,2 C_{\ell}\right)
\end{aligned}
##if not, is there a way to simplify the first line, i.e when adding the ##N## terms ##\chi^{2}## ?
2) I don't how to formulate correctly (on a level of vocabulary and notations) the notion of "equality in distribution", that is to say, is it correct to write :
##
\begin{aligned}
\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2} & \stackrel{d}{=} \sum_{\ell=1}^{N}(2 \ell+1) \chi^{2}\left(C_{\ell}\right)
\end{aligned}
##
?
The key point is the symbol ##\stackrel{d}{=}## : maybe I should rather write a sentence of kind :
The random variable ##\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}## follows the distribution equal to ##\sum_{\ell=1}^{N}(2 \ell+1) \chi^{2}\left(C_{\ell}\right)##
?
To give a more explicit example, if ##X_1## follows ##\text{pdf}_1## and ##X_2## follows ##\text{pdf}_2##, then Can I write ##X_1+X_2## following ##\text{pdf}_1+\text{pdf}_2## or even write directly :
##(X_1+X_2) \stackrel{d}{=} (\text{pdf}_1+\text{pdf}_2)## ?
or I saw also often in litterature :
##(X_1+X_2)## ~ (##\text{pdf}_1+\text{pdf}_2##)
I think this way of expression is bad and false (for example by taking 2 uniform laws). How to circumvent this problem of formulation ?As you can see, there are subtilities that I have not yet grasped.
Any help to better understand is welcome.
Important remark : ##C_{\ell}## depends on ##\ell## : ##C_{\ell} = C_{\ell}(\ell)##
\begin{aligned}
\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2} & \stackrel{d}{=} \sum_{\ell=1}^{N}(2 \ell+1) \chi^{2}\left(C_{\ell}\right) \\
& \stackrel{d}{=} \chi^{2}\left(\sum_{\ell=1}^{N}(2 \ell+1) C_{\ell}\right)
\end{aligned}
that is to say, include the sum on ##N## into ##\chi^2## ?
Maybe I could write instead (with shape/scale convention for Gamma distribution):
##
\begin{aligned}
\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2} & \stackrel{d}{=} \sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} \Gamma\left(1 / 2,2 C_{\ell}\right) \\
& \stackrel{d}{=} \sum_{\ell=1}^{N} \Gamma\left((2 \ell+1) / 2,2 C_{\ell}\right)
\end{aligned}
##if not, is there a way to simplify the first line, i.e when adding the ##N## terms ##\chi^{2}## ?
2) I don't how to formulate correctly (on a level of vocabulary and notations) the notion of "equality in distribution", that is to say, is it correct to write :
##
\begin{aligned}
\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2} & \stackrel{d}{=} \sum_{\ell=1}^{N}(2 \ell+1) \chi^{2}\left(C_{\ell}\right)
\end{aligned}
##
?
The key point is the symbol ##\stackrel{d}{=}## : maybe I should rather write a sentence of kind :
The random variable ##\sum_{\ell=1}^{N} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}## follows the distribution equal to ##\sum_{\ell=1}^{N}(2 \ell+1) \chi^{2}\left(C_{\ell}\right)##
?
To give a more explicit example, if ##X_1## follows ##\text{pdf}_1## and ##X_2## follows ##\text{pdf}_2##, then Can I write ##X_1+X_2## following ##\text{pdf}_1+\text{pdf}_2## or even write directly :
##(X_1+X_2) \stackrel{d}{=} (\text{pdf}_1+\text{pdf}_2)## ?
or I saw also often in litterature :
##(X_1+X_2)## ~ (##\text{pdf}_1+\text{pdf}_2##)
I think this way of expression is bad and false (for example by taking 2 uniform laws). How to circumvent this problem of formulation ?As you can see, there are subtilities that I have not yet grasped.
Any help to better understand is welcome.
Last edited: