Variance Sequences and Their Truncations

In the setting of The Real Numbers: Standing Notation and Background.

1. (Variance sequences)¶ A variance sequence is a sequence $c=(c_{k})_{k\in\mathbb{N}}$ of positive real numbers such that the series $\sum_{k=1}^{\infty}c_{k}$ converges.

2. (Truncations)¶ Let $c$ be a variance sequence and $n\in\mathbb{N}$. Read The Diagonal Gaussian Density on Euclidean Space and Its Notation and Diagonal Gaussian Measures on Euclidean Space, in the settings named there, with $n$ in place of the dimension written $d$ there. The truncation of $c$ to its first $n$ terms is $c^{(n)}=(c_{1},\dots,c_{n})$, a point of Euclidean space $\mathbb{R}^{n}$; it is a variance vector, since each of its entries is positive. The diagonal Gaussian measure on $\mathbb{R}^{n}$ with variances $c^{(n)}$ is written $\gamma_{c^{(n)}}$.