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Marchenko–Pastur law and sample-covariance edge bounds — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Analyticity of Holomorphic Functions; Liouville and Morera
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2 · Logical flowchart
3 · Definitions, theorems and proofs
None yet.
4 · Examples, counterexamples and false statements
Marchenko–Pastur law and edges below aspect one
Example
Assume AC. Let be one infinite real iid array with mean zero, variance one and finite fourth moment. Set and suppose . For , the limiting empirical spectral distribution has density and no atom at zero; the largest eigenvalue tends almost surely to , the smallest positive eigenvalue tends almost surely to , and has no zero eigenvalues eventually almost surely.
Example references
Facts & Assumptions
Given: AC, the entry-law assumptions in the Example, and .
The Axiom of Choice (The Axiom of Choice).
The Axiom of Choice implies countable choice (AC implies DC implies countable choice).
Under countable choice, the density, endpoints, and atom are defined by the Marchenko–Pastur formula (Marchenko–Pastur measure with aspect ratio gamma).
The MP weak law is stated in Almost-sure Marchenko–Pastur law for finite-fourth-moment sample covariance matrices.
The largest-edge conclusion is stated in Almost-sure limit of the largest sample-covariance eigenvalue.
The smallest-positive-edge and eventual-rank conclusions are stated in Almost-sure limit and structural-zero count at the lower sample-covariance edge.
Verification
Given: The hypotheses and notation in the Example.
Since , the endpoints are and , while the zero atom is . Substituting in [F1] gives the displayed density on and zero outside.
The nested array and AC satisfy [L1]–[L3]. The MP weak law [L1] gives the limiting density computed in step 1.1. Independently, [L2] gives largest edge , while [L3] gives smallest positive edge and eventual full column rank, hence no zero eigenvalues. The intersection of these probability-one events proves all assertions.
Structural zeros and the positive edge above aspect one
Example
Assume AC. Let be one infinite real iid array with mean zero, variance one and finite fourth moment. Set and suppose . For , the limiting empirical spectral distribution has density and an atom of mass at zero; almost surely has exactly zero eigenvalues eventually, its largest eigenvalue tends to , and its smallest positive eigenvalue tends to .
Example references
Facts & Assumptions
Given: AC, the entry-law assumptions in the Example, and .
The Axiom of Choice (The Axiom of Choice).
The Axiom of Choice implies countable choice (AC implies DC implies countable choice).
Under countable choice, the density, endpoints, and atom are defined by the Marchenko–Pastur formula (Marchenko–Pastur measure with aspect ratio gamma).
The MP weak law is stated in Almost-sure Marchenko–Pastur law for finite-fourth-moment sample covariance matrices.
The largest-edge conclusion is stated in Almost-sure limit of the largest sample-covariance eigenvalue.
The smallest-positive-edge and eventual-full-row-rank conclusions are stated in Almost-sure limit and structural-zero count at the lower sample-covariance edge.
Verification
Given: The hypotheses and notation in the Example.
Since , the endpoints are and , while the zero atom is . Substituting in [F1] gives the displayed density on and zero outside, plus the atom .
The nested array and AC satisfy [L1]–[L3]. The MP weak law [L1] gives the limiting density and atom computed in step 1.1. Independently, [L2] gives largest edge ; [L3] gives smallest positive edge and eventual full row rank, hence exactly zero eigenvalues. Intersecting these probability-one events proves the assertions.
The Marchenko–Pastur support and closing lower edge at aspect one
Example
Assume AC. Let be one infinite real iid array with mean zero, variance one and finite fourth moment. Set and suppose . For , the limiting empirical spectral distribution has density and no atom at zero; almost surely the largest eigenvalue tends to and the smallest positive eigenvalue tends to .
Example references
Facts & Assumptions
Given: AC, the entry-law assumptions in the Example, and .
The Axiom of Choice (The Axiom of Choice).
The Axiom of Choice implies countable choice (AC implies DC implies countable choice).
Under countable choice, the density, endpoints, and atom are defined by the Marchenko–Pastur formula (Marchenko–Pastur measure with aspect ratio gamma).
The MP weak law is stated in Almost-sure Marchenko–Pastur law for finite-fourth-moment sample covariance matrices.
The largest-edge conclusion is stated in Almost-sure limit of the largest sample-covariance eigenvalue.
The smallest-positive-edge conclusion is stated in Almost-sure limit and structural-zero count at the lower sample-covariance edge.
Verification
Given: The hypotheses and notation in the Example.
At , the endpoints are and , and the atom is zero. On the density simplifies as , giving the formula in the Example.
The nested array and AC satisfy [L1]–[L3]. The MP weak law [L1] gives the density computed in step 1.1. Independently, [L2] gives largest edge , while [L3] gives an eventually defined smallest positive eigenvalue tending to zero. Their simultaneous probability-one conclusions prove the Example, without inferring either edge from the weak law.