Let $v = (v_1,ldots,v_n)$ and $(w_{1,1},ldots,w_{1,n},ldots, w_{n,m})$ be random vectors with iid coordinates, and also $v$ is independent of $w$, with $w_{i,j} sim N(0,1/m)$ and $v_j sim N(0,1/n)$, for $i=1,ldots,n$, $j=1,ldots,m$. Define a random function $f_{w,v}:mathbb R^m to mathbb R$ by $f_{w,v}(x) := sum_{i=1}^nv_iphi(sum_{j=1}^m w_{i,j}x_j)$, where $phi(t):=max(t,0)$. Fix a point $a, b in mathbb R^m$.

**Question.** How to got about computing

$$

p_{m,n}(a,b) := mathbb P(f_{w,v}(x)f_{w,v}(a, b) > 0),

$$

Or even just $lim_{n to infty}lim_{m to infty}p_{m,n}(a,b)$.

Note that $p_{n,m}(a,b)$ is simply the probability that the random field $x mapsto f_{w,v}(x)$ flips its sign between $x=a$ and $x=b$.

## Observations

- Conditioned on $w$, we compute (see this math.SE post)

$$mathbb P(f_{w,v}(x)f_{w,v}(a, b) > 0) = kappa_0(phi_w(a),phi_w(b))

$$

where $mathbb R^n ni phi_w(x) := (phi(sum_{j=1}^mw_{ij}x_j))_{1 le i le n}$, and $kappa_0(z,z’) := 1-frac{1}{pi}arccos(z^Tz’/|z||z’|) in (0, 1)$ is the **arc-cosine kernel** of order $0$.
- Thus, we have the identity

$$

p_{m,n}(a,b) = mathbb E_w(kappa_0(phi_w(a),phi_w(b))).tag{2}

$$

**Question.** Can the formula (2) be further simplified ? Is it linked to some other kernels ?