Limiting characteristic function of the total solvency shock

Let $S^{(0)}_{1}=\sum_{w\ne 1}S^{(0)}_{w1}$ be the total solvency shock transmitted to bank $1$ in step $0$, conditioned on the type $T_1=T$. Let $\hat f^{(0)}_\Delta(k'\mid T')$, $\mathbb{P}(T')$, and $\kappa(T',T)$ denote the quantities associated with the type-conditioned individual shocks and the inhomogeneous random financial network, and define

$$R(k,k'\mid T,T'):=\frac{1}{2\pi}\int_{-\infty}^{0}e^{ik'y}R(k,-y/\lambda)\,\mathrm{d}y,$$

where $R(k,a)$ is given by the source's formula with $N=\infty$ and conditions $T,T'$. Total-solvency-shock characteristic-function conjecture. The characteristic function has the limiting behaviour

$$\mathbb{E}^{(N)}[e^{ikS^{(0)}_{1}}\mid T]=\hat f^{(0)}_{S}(k\mid T)(1+O(N^{-1})),$$

with

$$\hat f^{(0)}_{S}(k\mid T):=\exp\left(\sum_{T'}\mathbb{P}(T')\kappa(T',T)\int_{-\infty}^{\infty}\hat f^{(0)}_\Delta(k'\mid T')R(k,k'\mid T,T')\,\mathrm{d}k'\right),$$

where the limit of the logarithm is in $L^2[0,\infty)$. This conjecture results from an unproved asymptotic-independence step for the full collection of shocks, beyond the finite-collection result established in the surrounding discussion; rigorous proof remains open.