Notation and conventions
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All vector spaces and matrices in this book are finite dimensional unless stated otherwise. Matrices and factors are real, while transform arguments may be complex. Gaussian components in a spectral reduction are independent standard normal variables. Logarithms are natural unless the base is displayed.
| Symbol | Meaning |
|---|---|
| \(A\) | A factor. In the streaming construction, \(A\in\mathbb R^{n\times m}\). |
| \(K=AA^\top\) | The sample-space Gram matrix. |
| \(H=A^\top A\) | The feature-space Gram matrix in the phase construction. Elsewhere \(H\) denotes the explicitly defined symmetric coefficient matrix. |
| \(Q\) | The scalar Gaussian quadratic form whose law is being evaluated. |
| \(\lambda_j\) | Its nonzero spectral coefficients, including multiplicity. |
| \(q\) | The threshold in \(P(Q\ge q)\). |
| \(\kappa(s)\) | The cumulant generating function, \(\log\mathbb E e^{sQ}\). |
| \(\theta\) | A positive tilt in the moment-generating-function domain. |
| \(b\), \(N\) | The upper support endpoint and the number of retained COS modes. |
| \(u_k\), \(J_k\) | Frequency \(k\pi/b\) and its exponentially weighted payoff integral. |
| \(\|\cdot\|_2\) | Euclidean vector norm or induced matrix norm, as appropriate. |
| \(\|\cdot\|_F\) | Frobenius norm. |
| \(M\succeq0\) | Positive semidefiniteness. |
The symbols \(U\), \(B\), and \(R\) have locally defined linear-algebra roles. A core or basis is never assumed to span the exact range unless that hypothesis is stated. Nonzero spectra are compared with multiplicity. Additional zero eigenvalues in an oversized core do not change the quadratic-form law.
The implementation retains some earlier variable names. Appendix C maps those names to the notation used here. Printed endpoint strings, not rounded display fields, are the numerical record of an enclosure.