Suppose you have the following sequences of statistically independent Gaussian random variables with zero means and variances \(\sigma^{2}\) if

\[
X_{1}, X_{2}, \ldots, X_{N} ; X_{i}=A_{i} \cos \Theta_{i} \text { and } Y_{1}, Y_{2}, \ldots, Y_{N} ; Y_{i}=A_{i} \sin \Theta_{i}
\]

Define \(Z=\sum_{i=1}^{N} A_{i}^{2}\). Find an expression where \(Z\) exceeds a threshold value \(v_{T}\).