I want to know how the come up with the density of $X^{2}_{t}=W^{2}_{1}(t)+W^{2}_{2}(t)+W^{2}_{3}(t)$.
I know that I need to prove it by considering 2 cases: 1. $X_{t}$ starts at $x>0$; 2. $X_{t}$ starts at $x=0$. Case 2 can be derived easily. However, I am not sure how to derive case 1.
When I derive case 1, I try to find the density $f_{X^{2}_t}(y;x)$ first, which ended up with a term involving modified Bessel function of first kind $I_{\nu}(z)$ as follow: $$\frac{1}{2t}(\frac{y}{x})^{-1/2}\exp(-\frac{x+y}{2t})I_{1}(\frac{\sqrt{xy}}{t})$$ Unfortunately, I do not know how to recover to the density $f_{X_t}(y;x)$, which should be as follow: $$\frac{y}{x}\frac{1}{\sqrt{2\pi t}}\{\exp(-\frac{(x-y)^2}{2t})-\exp(-\frac{(x+y)^2}{2t})\} $$ Could anyone provide hints on changing it?