Suppose $$C_n=X_1 X_2\cdots X_n$$ where $X_i$ is $d\times d$ matrix with IID entries normally distributed with mean 0 and variance $\frac{1}{d}$, 

The following appears to be true for large $d$, why?

$$\|C_n C_n^T\|_F^2\approx d(n+1)$$

Here are some numbers from a 4 samples with $d=1000$

$$\left(
\begin{array}{cccccc}
  & \text{n} & \text{sample1} & \text{sample2} & \text{sample3} & \text{sample4} \\
  & 1 & 2003.99 & 1998.66 & 1999.51 & 1998.14 \\
  & 2 & 3029.97 & 2990.12 & 3008.21 & 2999.13 \\
  & 3 & 3967.81 & 3995.46 & 4022.33 & 4005.2 \\
  & 4 & 5027.41 & 5075.39 & 4941.94 & 5057.4 \\
  & 5 & 6143.21 & 5964.35 & 5844.76 & 6015.08 \\
\end{array}
\right)$$

[Notebook](https://www.wolframcloud.com/obj/yaroslavvb/nn-linear/forum-growth-of-traces.nb)