If we use the integral expression for the \( \delta \)-function
$$ \delta(z-\frac{x_1+x_2+\dots+x_m}{m})=\frac{1}{2\pi}\int_{-\infty}^{\infty} dq\exp{\left(iq(z-\frac{x_1+x_2+\dots+x_m}{m})\right)}, $$and inserting \( e^{i\mu q-i\mu q} \) where \( \mu \) is the mean value we arrive at
$$ \tilde{p}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty} dq\exp{\left(iq(z-\mu)\right)}\left[\int_{-\infty}^{\infty} dxp(x)\exp{\left(iq(\mu-x)/m\right)}\right]^m, $$with the integral over \( x \) resulting in
$$ \int_{-\infty}^{\infty}dxp(x)\exp{\left(iq(\mu-x)/m\right)}= \int_{-\infty}^{\infty}dxp(x) \left[1+\frac{iq(\mu-x)}{m}-\frac{q^2(\mu-x)^2}{2m^2}+\dots\right]. $$