Suppose that we have random variables $X_1$, $X_2$ each drawn independently from Irwin Hall Distribution with same mean of 0 and different variances.
If I have $B_1$ and $B_2$, which are high probability upper bound of $X_1$ and $X_2$ respectively, can I presume the upper bound of $X_1 + X_2$?
In other words,
If $Pr[X_1 \gt B_1] \lt \epsilon$, $Pr[X_2 \gt B_2] \lt \epsilon$ holds with negligible probability $\epsilon$, Can I find the tight upper bound $B$ s.t. $Pr[X_1 + X_2 \gt B] \lt \epsilon$?
I thought of two candidates for upper bound B.
One is $B_1 + B_2$ and the other is $\sqrt{B_1^2+B_2^2}$.
I thought that first candidate must satisfy the upper bound of $X_1 + X_2$, But it would be a loose bound.
Second one would be better and tight upper bound, but I can't prove $Pr[X_1 + X_2 \gt \sqrt{B_1^2+B_2^2}] \lt \epsilon$
I got idea from variance of sum of random variables.
If $X_1$ ~ $(0, \sigma_1^2)$ , $X_2$ ~ $(0, \sigma_2^2)$, $X_1 + X_2$ will follow the distribution of $(0, \sigma_1^2 + \sigma_2^2)$
Is this thought reasonable?
More generally, I also want to know the upper bound of $X_1$, $X_2$, … , $X_n$.
If the second candidate is true, I would like to know how to prove it.
Thank you for reading.
Best Answer
The event $X_1+X_2 > B_1+B_2$ is contained in the union of the two events $X_1>B_1$ and $X_2>B_2$. (This assertion is equivalent to the algebraic implication "if $X_1+X_2>B_1+B_2$ then $X_1>B_1$ or $X_2>B_2$", which is equivalent to the easily verified "if $X_1\le B_1$ and $X_2\le B_2$ then $X_1+X_2\le B_1+B_2$".)
In particular, if $Pr[X_1>B_1]<\varepsilon$ and $Pr[X_2>B_2]<\varepsilon$, then $$ Pr[X_1+X_2>B_1+B_2] \le Pr[X_1>B_1] + Pr[X_2>B_2] < 2\varepsilon. $$ It's not hard to show by example that this bound is best possible: let $X_j$ take only values $B_j-1$ and $B_j+2$ with probabilities $1-\delta$ and $\delta$ where $\delta$ is just less than $\varepsilon$, anticorrelated so that $X_1=B_1+2$ and $X_2=B_2+2$ never occur simultaneously.