The original submitted PDF is preserved unchanged. This separate note addresses three specific points and clarifies the scope of the work. Page references below use the thesis’s printed numbering; each link opens the corresponding PDF page.
1. Positive definite and semidefinite
Printed page 21 · Definition 3.3
The definition labeled “Positive Definite Matrix” uses a nonnegative quadratic-form condition. For a real symmetric matrix K, that condition defines positive semidefiniteness:
zTKz ≥ 0for every z ∈ ℝn.
Positive definiteness requires a strict inequality for every nonzero vector:
zTKz > 0for every nonzero z ∈ ℝn.
For the standard definitions, see Boyd and Vandenberghe, Convex Optimization, Appendix A.5.2, printed page 647.
2. Positive entries are insufficient
Printed page 21 · Paragraph before Definition 3.2
The paragraph infers positive definiteness from symmetry and positive matrix entries. Those properties alone do not imply positive definiteness. For example:
K = [[1, 2], [2, 1]],z = (1, −1)T.
This matrix is symmetric and all its entries are positive, yet:
zTKz = 1 − 2 − 2 + 1 = −2.
The counterexample identifies a gap in the stated argument. It does not disprove positive definiteness of Gaussian kernel matrices; that property requires a kernel-specific justification with the appropriate assumptions.
3. Normalize weights to one
Printed page 28 · “Bi-stochastic Matrix with Sinkhorn”
The statement that the nonnegative weights “add to zero row wise” should instead require a sum of one. Normalized mixture weights sum to one so that the resulting mixture of densities integrates to one.
In the displayed definition of Zi,k, k is fixed and the normalization sums over the sample index i:
∑i=1N Zi,k = 1for every k.
With this indexing, these are column sums. A bi-stochastic matrix has both row and column sums equal to one, consistent with the two conditions displayed in Definition 3.7.
Scope of the original work
The thesis develops a preliminary approach connecting distributional forecasting and sequential financial decisions. On printed page 23, it states that the conditional kernel densities for the stocks had not yet been computed. The conclusion on printed page 32 describes the trading application and its empirical evaluation as future work.
The submitted work therefore does not establish a completed forecasting or trading system, validated investment performance, or outperformance. The interactive Gaussian transport illustration on this website is a separate educational demonstration.