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dohmatob
dohmatob
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Limiting distribution of "scatter matrix" $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ as $n \rightarrow \infty$ for iid $x_1,\ldots,x_n \in \mathbb R^p$

Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s.

Question. What is the imitinglimiting distribution of the $p$-by-$p$ psd matrix"scatter matrix" $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ as $n \rightarrow \infty$ ?

Observations

  • If the $x_i$'s are from a centered multivariate Gaussian, then $XX^T$ has a Wishart distribution.

Limiting distribution of $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ as $n \rightarrow \infty$ for iid $x_1,\ldots,x_n \in \mathbb R^p$

Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s.

Question. What is the imiting distribution of the $p$-by-$p$ psd matrix $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ as $n \rightarrow \infty$ ?

Observations

  • If the $x_i$'s are from a centered multivariate Gaussian, then $XX^T$ has a Wishart distribution.

Limiting distribution of "scatter matrix" $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ for iid $x_1,\ldots,x_n \in \mathbb R^p$

Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s.

Question. What is the limiting distribution of the "scatter matrix" $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ as $n \rightarrow \infty$ ?

Observations

  • If the $x_i$'s are from a centered multivariate Gaussian, then $XX^T$ has a Wishart distribution.
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dohmatob
dohmatob
  • 6.9k
  • 1
  • 18
  • 76

Limiting distribution of $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ as $n \rightarrow \infty$ for iid $x_1,\ldots,x_n \in \mathbb R^p$

Let $x_1,\ldots,x_n$ be drawn iid from such "nice" distribution on $\mathbb R^p$ (but possibly very general!), and let $X$ be the $n$-by-$p$ matrix formed by vertically stacking the $x_i$'s.

Question. What is the imiting distribution of the $p$-by-$p$ psd matrix $\frac{1}{n}XX^T:=\frac{1}{n}\sum_{i=1}^nx_ix_i^T$ as $n \rightarrow \infty$ ?

Observations

  • If the $x_i$'s are from a centered multivariate Gaussian, then $XX^T$ has a Wishart distribution.