# False negative probability of cross-correlation

Suppose $$D$$ is a string of length $$L$$ where its values are uniformly distributed in $$[-a,a]$$. Also, all values of $$D$$ are independent (i.i.d).

$$X$$ is a noisy version of $$D$$ in this way: $$X=D+N$$ where $$N$$ is a white Gaussian noise with mean and variance of $$0$$ and $$\sigma_N^2$$.

$$P$$ is a sub-string of $$D$$ where its length is $$l$$ $$(l.

To detect $$P$$ in $$X$$, I use the cross-correlation of $$X$$ and $$P$$: $$C_n = \sum_{i=1}^l X_{n+i-1}P_i \space\space , \space\space 1 and a threshold of $$T=K\sigma_{C_n}$$ where $$K$$ is a constant value and $$\sigma_{C_n}$$ is the standard deviation of the correlation signal $$C_n$$.

A false negative occurs If no points of the correlation signal reaches the threshold.

Now the question is: how can I theoretically calculate the false negative probability? It tried to use Q-function, but I couldn't achieve the answer.