# Is it that trivial to see that a sequence of random variables is mutually independent?

Grinstead and Snells book, Introduction to Probability, page 144:

0) The authors say that they consider "special classes of random variables", one such classe being the class of indepedent trails. I think this is imprecise: They should have said "classes of sequences of random variables". What do you think ?

1) The $X_j$ are functions $X_j:R\times R\times \ldots \times R \rightarrow \mathbb{R}$, right ?

2) They should have specified that $R\subseteq \mathbb{R}$ since otherwise the $j$-th projection isn't well defined: $X_j(\Omega)\subseteq \mathbb{R}$, but it $R\ni \omega_j \not\in \mathbb{R}$.

3) On the second line from below shouldn't it say "outcome $(\omega_1,\ldots,\omega_n)$, rather then $(r_1,\ldots,r_n)$ ? (The $r$'s are also already used to define $R=\{r_1,\ldots,r_s\}$)

4) Most important Is it that trivial to see (penultimate line) that the random variables $X_1,\ldots,X_n$ form an independent trials process ? It is indeed easy to see, that they have the same distribution, but proving that they are mutually independent does require some work!

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0) That you are formally right. 1) No, $X_j:R^n\to R$ and in fact, $X_j$ is the projection on the $j$th coordinate. 2) No, see above. 3) Indeed it should read $(\omega_j)_{1\leqslant j\leqslant n}$ with each $\omega_j$ in $R$ since the authors took care to explain that $\omega$ is the running element of $\Omega=R^n$. 4) The answer depends on your definition of trivial but that the result holds should be clear to anybody determined to check it for themselves with a pen and a sheet of paper.
Uhm...to what would you suggest that I should change my title to ?  I also have some unclarities: To 1) Isn't it customary (as long as we aren't in "high dimensions" of probability and measure theory) to let a (discrete) random variable be real-valued ? For example here (first link in google) stat.yale.edu/Courses/1997-98/101/ranvar.htm they say that random variable usually take numerical values; or in wikipedia where it's also noted that it's usual real-valued and that other types of "random variables" are called "random elements".(cont.) – MyCatsHat Dec 30 '12 at 16:20
(cont.) I mean, it seems odd that an intro text should stray away from what seems to be mainstream and usual wording. 3) So you mean it's ok in book ? In that case I don't understand it...it seems to me that something has to be wrong with the indexes: $r_i$'s, with indexes $\{1,\ldots,s\}$ are already used in the definition of $R$. How can they be used again with different indexes ranging $\{1,\ldots,n\}$?  4) Trivial means to me that it's a one-liner; I checked it and it definitely wasn't a one-liner (it's true that I was very explicit and wrote everything out). (cont.) – MyCatsHat Dec 30 '12 at 16:20
Title: your problem, not mine (but cancelling everything before "OR" would be fine). Re 1) and customs: there is no disadvantages to calling random variables measurable functions with values in any measurable space $(E,\mathcal E)$ (and I am not sure that what you call mainstream is that mainstream). Re 3): you were right from the start, I misread. Re 4): nothing to add nor to delete from my answer. – Did Dec 30 '12 at 21:53