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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

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 4 samples with $d=1000$ (Notebook):

n     sample1  sample2  sample3  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  4005.20
4     5027.41  5075.39  4941.94  5057.40
5     6143.21  5964.35  5844.76  6015.08

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 $\frac{1}{d}$,

Why is the following true for large $d$?

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

Here are some numbers from a single sample with $d=1000$

$$\left( \begin{array}{ccc} & \text{n} & \text{norm} \\ & 1 & 2004.09 \\ & 2 & 2998.89 \\ & 3 & 4008.74 \\ & 4 & 4946.42 \\ & 5 & 6009.35 \\ \end{array} \right)$$

Notebook

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