6

Unfortunately, this turns out to be less interesting than I thought. After a short comment of a collegue I found out that this convergence phenomenon is just a special case of the following easy statement: Let $V_n$ and $V$ be random variables with $V_n \to V$ in distribution. Let $K$ be a stochastic kernel with the $C_b$-Feller property (i.e. $ x \mapsto ...


4

We generally call it the Closed Form of the expression . So $\dfrac{n(n+1)}{2}$ is the closed form of $\sum_{k=1}^n k$ .


2

Population standard deviation (i.e. where you divide by the sample size) will tell you the standard deviation of the heights of your class considered as a whole, i.e. you don't care about the fact that people from other classes or other parts of the world might have different heights. Sample standard deviation (where you divide by the sample size minus $1$) ...


1

As a commenter noted, this isn't really a technical usage. What they are saying is that, whenever we grow the tree another step, the error will get smaller. However as we start to overfit, the error will generally go down less than it did on earlier steps. So we set a threshold (say $0.1$) that says "if the error only goes down by $0.1$ or less, we stop ...


1

Ronald Brown, Topology and Groupoids (2006), p.225: Let $X, Y$ be topological spaces. A map $F \colon X \times [0, q] \to Y$ will be called a homotopy of length $q;$ for such $F,$ the initial map and the final map of $F$ are respectively the functions $$ \begin{aligned} f \colon X & \to Y \\ x & \mapsto F(x, 0) \end{aligned} \qquad \begin{...


1

You will find the definition within the statement of Theorem 7 of the linked paper: An almost $\sigma$-bounded set is the union of a $\sigma$-bounded set and a null set. A $\sigma$-bounded set is (as per the definition given in the peliminary section, compare @D.R 's answer) the union of countably many bounded sets.


1

On Wikipedia it's called a semiring of sets also in my measure theory texts, though the Germans seem to prefer Semialgebra, which why your surname (Lenz) might be the reason you come at it from this name.


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