Let
$$D=\frac{1}{N}\sum_{n=1}^{N}H\left(\xi_{n}-1\right),$$
such that $\xi$ denotes a Gaussian random walk with mean $\mu$ and $\sigma$, passed through the Heaviside function
$$H(x-1)=\begin{cases} 1, & x>1\\ 0, & x\leq1. \end{cases}$$
Can any statements be made about the family of distributions $H(\xi_n-1)$ belongs to, or its mean and variance? Any help would be much appreciated. Please see answer to this question below where a new question is posed.
Realisation of Random Walk (for $N=1000$)
Red line $D$, black line is $\xi_n$ (with $\mu=1$ and $\sigma=0.16$), and purple line is $H(\xi_n-1)$.
For positive $\sigma$,
$$ \begin{split} D&=\mathbb{E}\left[H\left(\xi_{n}-1\right)\right],\\&=\mathbb{P}\left(\xi_{n}\geq1\right),\\&\sim\int_{1}^{\infty}\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{\ell-\mu}{\sigma}\right)^{2}\right)d\ell,\\&\sim\frac{1}{2}\left(1-\text{erf}\left(\frac{1-\mu}{\sigma\sqrt{2}}\right)\right). \end{split}$$
The $\sim$ is used to denote asymptotic equivalence. So, for infinite $n$ the relation holds since $$\frac{1}{2}\left(1-\text{erf}\left(\frac{1-\mu}{\sigma\sqrt{2}}\right)\right)=\mathbb{E}\left[\mathbb{P}\left(\xi_{n}\geq1\right)\right].$$ My question may be updated to what is the error term for the approximation? So, can $g(N)$ be found in $$\mathbb{P}\left(\xi_{n}\geq1\right)=\frac{1}{2}\left(1-\text{erf}\left(\frac{1-\mu}{\sigma\sqrt{2}}\right)\right)+\mathcal{O}(g(N))?$$