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Stieltjes transform of a probability measure on the real line
Definition
Let be a Borel probability measure on . For with and , its Stieltjes transform, with the convention, is The integral is finite because for every real .
Definition references
Bai and Silverstein, Appendix B.2, printed pp. 514–523 (complete book PDF).
Junyi Liao, §1.3.1, printed pp. 17–18 (complete lecture notes).
Depends on
Used by
- An anisotropic population with a balanced deterministic direction Example
- Deterministic equivalent for the bounded complex sample-covariance resolvent Lemma
- Stieltjes inversion and uniqueness for probability measures on the real line Lemma
- Almost-sure Marchenko–Pastur law for finite-fourth-moment sample covariance matrices Theorem
- Directional deterministic equivalent for bounded sample-covariance entries Theorem
- General-population sample-covariance law for bounded entries Theorem
- Sample-covariance resolvent deterministic equivalents Theorem
Dependency tree · two levels
7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Bai and Silverstein, Spectral Analysis of Large Dimensional Random Matrices, 2nd ed., Appendix B.2 (standard reference, not scraped)
- Junyi Liao, Random Matrix Theory, §1.3.1 (standard reference, not scraped)