The joint probability of moving sums There are $N$ independent random variables $k_1,k_2, ...,k_N$ which have the same Normal distribution $N(\mu,\sigma^2)$.
We define the sequence of sums of length $s$:
$S_1=k_1+k_2+...+k_s$,
$S_2=k_2+k_3+...+k_{s+1},$
$S_3= ...$
How can I find the probability that all sums exceed a real number $M$ at the same time: $P(S_1>M,S_2>M,...,S_{N-s+1}>M)$?
 A: Now I can partially answer your question. The solution you are looking for might look like the following:
Since each random variable distributed normally and they are independent of each other, we have:
$k_1 + k_2 + ... + k_s = S_1 \sim N(s\mu, s\sigma^2)$
$k_2 + k_2 + ... + k_{s+1} = S_2 \sim N(s\mu, s\sigma^2)$
Since you need to calculate joint probability, we need to look at the joint distribution:
Lets define:
$\begin{align}
    \vec{S} &= \begin{bmatrix}
           S_{1} \\
           S_{2} \\
           \vdots \\
           S_{N-s+1}
         \end{bmatrix}
  \end{align}$,
$\begin{align}
    \vec{\theta} &= \begin{bmatrix}
           s\mu \\
           s\mu \\
           \vdots \\
           s\mu
         \end{bmatrix}
  \end{align}$,
$\begin{align}
    \Sigma_S &= \begin{bmatrix}
           s\sigma^2 & \sigma_{S_1, S_2} & ... & \sigma_{S_1, S_{N-s+1}} \\
           \sigma_{S_2, S_1} & s\sigma^2 & ... &  \sigma_{S_2, S_{N-s+1}}\\
           \vdots \\
           \sigma_{S_{N-s+1}, S_1} & \sigma_{S_{N-s+1}, S_2} & ... & s\sigma^2
         \end{bmatrix}
  \end{align}$
Where:
$\sigma_{S_k, S_z} = Cov(S_k, S_z)$
Then we have:
$$\vec{S} \sim N(\vec{\theta}, \Sigma_s)$$
From here you need to calculate: $P(\vec{S} > M) = 1 - P(\vec{S} \leq M) = 1 - F_{\vec{S}}(M)$
Which is the CDF of the multi-variate normal distrb. Unfortunately, there is no close form solution, but there exist some numerical and approximate methods to calculate it.
see this R library for numerical methods: https://github.com/lbelzile/TruncatedNormal
see this document for approximation: https://upload.wikimedia.org/wikipedia/commons/a/a2/Cumulative_function_n_dimensional_Gaussians_12.2013.pdf
