Help understanding the definition of positive semidefinite matrix

[docnet]

Homework Statement

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Relevant Equations

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Please confirm or deny the correctness of my understanding about this definition.

For a given set of $t_i$s, the matrix $(B(t_i,t_j))^k_{i,j=1}$ is a constant $k\times k$ matrix, whose entries are given by $B(t_i,t_j)$ for each $i$ and $j$.

The the 'finite' in the last line of the definition refers to $t_1$ and $t_k$ is finite, and $k$ is assumed to be a finite integer.

And if we impose the condition $t_1<t_2<...<t_k$ , then for all finite time slices' $\{t_i\}_{i=1}^k$ means $\{t_1,...,t_k | (t_1<t_2<...<t_k) \text{ and } (t_i \in \mathbb{R} \text{ for all } i \in \{1,...,k\}) \text{ and } (-\infty < t_1 < t_k < \infty)\}.$

One such ' time slice' is $1,2,3,...k$. Another is $-1,-,\frac{1}{2},...,-\frac{1}{k-1},-\frac{1}{k}.$


A few questions I have are, what information does $\{t_i\}^k_{i=1}$ convey? I find interpreting this notation confusing. Is 'time slice' a precise term at all?

Thank you.

 

[docnet]

To add, if someone defines the function $B$ as $B(s,t)=\mathbb{E}[X_sX_t]$, the matrix $((B(i,j))_{i,j=1}^k)=M$ is symmetric, i.e., $M_{ij}=M_{ji}=\mathbb{E}[X_sX_t]= \mathbb{E}[X_tX_s].$