When ray tracing, I find it intuitively clear that deeper paths have a lower contribution to the overal picture than shorter paths. This is the main reason why it is generally okay to render a picture only to a certain depth, but I am currently not able to prove this.
Formally, we can write the rendering equation as
\begin{align*} L_o(\vec{p}_0, \omega_0) &= L_e(\vec{p}_0, \omega_0) \\ &+ \int_{\Omega} f_r(\vec{p}_0, \omega_0, -\omega_1) \cos{\theta_1} L_e(\vec{p}_1, \omega_1) \text{d} \omega_1 \\ &+ \int_{\Omega^2} f_r(\vec{p}_0, \omega_0, -\omega_1) \cos{\theta_1} f_r(\vec{p}_1, \omega_1, -\omega_2) \cos{\theta_2} L_e(\vec{p}_2, \omega_2) \text{d} \omega_1 \text{d} \omega_2 \\ &+ \dots \,, \end{align*} where $\vec{p}_{j+1}$ is defined as the intersection of a ray with origin $\vec{p}_j$ and direction $-\omega_{j+1}$ and $\theta_{j+1}$ is the angle between the normal of the surface at $\vec{p}_j$ and $-\omega_{j+1}$. Each term represents the radiance contribution of respectively paths with depth 0, 1, 2, and so on. Let's denote the contribution of depth $d$ with $L_o^{(d)}(\vec{p}_0, \omega_0)$. I suspect that, under mild conditions on the scene, the contributions of these terms will asymptotically decrease exponentially: $$L_o^{(d)}(\vec{p}_0, \omega_0) = \mathcal{O}(t^d) \,\text{ as } d \rightarrow \infty$$ for some $t$ with $0 < t < 1$, where I have used the big-O notation.
Of course, I expect this only to hold when the scene is somehow "realistic", which I define as follows.
Firstly, all light sources should have only a limited amount of power. Note that infinite radiances, like with point lights, should be permitted, as long as their power is finite.
- The total power of all light sources should be finite.

Secondly, we should exclude scenes that consists only of mirrors or diffuse white materials, or any other material that perfectly conserves energy. In other words, the scene should always absorb a little energy. These materials should still be permitted though, as long as their reflected light eventually ends up in materials that do not conserve the energy. I tried to formalize this the following way.
- There exists an integer $k$ with the following property: for every point $\vec{p}$ in the scene there should be a path of length $k$ that starts in $\vec{p}$ and which has a throughput smaller than 1.

Questions.
Using these assumptions, is it possible to prove my suspicion? How would you do it?
If it isn't possible, are there other "realistic" conditions that will make it proveable? Any conditions you come up with should not limit scenes that are used in practice in any way. (I know, sometimes people create materials that transmit more light than what comes in. You can ignore these materials).
What I tried so far.
I was able to prove this with some stronger conditions, namely
- There is an upper bound $L_{max}$ on emitted radiance $L_e$.
- There is an upper bound $t < 1$ on the percentage of energy that materials conserve.

With these conditions, the prove is straightforward, but they do not allow point lights or perfectly white diffuse materials and such.
Proof. \begin{align*} L_o^{(d)}(\vec{p}_0, \omega_0) &= \int_{\Omega^d} \left(\prod_{j=1}^d f_r(\vec{p}_{j-1}, \omega_{j-1}, -\omega_j) \cos{\theta_j} \right) L_e(\vec{p}_d, \omega_d) \text{d} \omega_1 \dots \text{d} \omega_d \end{align*} Using condition (1): \begin{align*} &\leq L_{max} \int_{\Omega^d} \prod_{j=1}^d f_r(\vec{p}_{j-1}, \omega_{j-1}, -\omega_j) \cos{\theta_j} \text{d}\omega_j \\ &= L_{max} \int_{\Omega} \text{d}\omega_1 f_r(\vec{p}_{0}, \omega_{0}, -\omega_1) \cos{\theta_1} \int_\Omega \dotsm \int_{\Omega} \text{d}\omega_d f_r(\vec{p}_{d-1}, \omega_{d-1}, -\omega_d) \cos{\theta_d} \end{align*} By applying condition (2) recursively on the innermost integral: \begin{align*} &\leq L_{max} t^d \end{align*}