I am given a sequence $X=\{X_1,X_2,\ldots,X_n\}$ of $n$ i.i.d. zero-mean Gaussian random variables $X_i\sim\mathcal{N}(0,\sigma^2)$, and a vector $\mathbf{y}=\{y_1, y_2, \ldots, y_m\}$ of $m$ real numbers, where $m\leq n$. I am trying to write down an expression for the probability $p$ of occurrence of a sub-sequence $X_S=\{X_i,\ldots,X_{i+m-1}\}$ in sequence $X$ such that the point it represents in the $m$-dimensional Euclidean space is closer (in Euclidean distance) to $\mathbf{y}$ as opposed to the origin.
I know that, due to the rotational symmetry of the Gaussian distribution, the probability of a point formed $m$ i.i.d. zero-mean Gaussian random variables being closer to $y$ can be expressed as $1-\Phi\left(\frac{\|\mathbf{y}\|}{2\sigma}\right)$, where $\|\cdot\|$ is the Euclidean norm and $\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^xe^{-t^2}dt$ is the cumulative distribution function of a standard Gaussian random variable. I can thus use the union bound to upper-bound the probability that I am looking for as follows:
$$p\leq (n-m)\left(1-\Phi\left(\frac{\|\mathbf{y}\|}{2\sigma}\right)\right)$$
However, this seems awfully loose and I wonder if an exact expression is available.