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Results tagged with measure-theory
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A field of mathematics concerning measures, which are functions mapping sets to real numbers that generalize the idea of area and volume. Measure theory underlies modern treatments of probability. Questions about measure theory with a weak connection to probability or statistics may be more suitable for math.stackexchange.com than this site.
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How to prove $\int_{\mathbb{R}} g(x) dF^n(x) = n \int_{\mathbb{R}} g(x) F^{n-1}(x) dF(x)$
Let $F$ be a distribution function, and let $g \colon \mathbb{R} \to \mathbb{R}$ be a real function.
I want to prove $\int_{\mathbb{R}} g(x) dF^n(x) = n \int_{\mathbb{R}} g(x) F^{n-1}(x) dF(x)$, wher …
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Equivalence of Tightness of Seqeuence of CDFs
In Durrett, a sequence of cdf $\{F_n\}$ is called tight if for all $\epsilon > 0$< there exists $M_\epsilon$ such that $\limsup 1-F_n(M_\epsilon)+F_n(-M_\epsilon) \leq \epsilon$.
In Rosenthal, a seque …
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$1-F$ is rapidly varying if and only if there exists $b_n$ such that $\frac{\max X_i}{b_n} \...
The following is a problem from Extreme Values, Regular Variation and Point Processes by Resnick.
We will say $1-F$ is rapidly varying as $x \to \infty$ if $\lim_{t \to \infty} \frac{1-F(tx)}{1-F(t)} …
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Definition of exponent measure (extreme value theory)
Let $F$ be a distribution function on $\mathbb{R}^2$, and let $U_i$ be the left continuous inverse of $\frac{1}{1-F_i}$, where $F_i$ is the marginal distribution of $F$.
In my textbook, there is the f …
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How to (dis)prove $\lim_{k\to\infty}\lim_{n\to\infty}E(Y_{n,K}) = E(\min(X_n, K))$?
Here is the problem:
Given $X, X_1, X_2, \ldots$, non-negative random variable with finite expectation and $X_n \to X$ pointwise and $Y_{n,K} = \min(X_n,K)$, we are asked to see if
a) $\lim_{K \to \i …
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Density from characteristic function: Durrett example 3.3.8 and 3.3.9
Letting $\varphi(t)$ be the characteristic function for the probability measure $\mu$, we know if $\int \left|\varphi(t)\right|dt < \infty$, then $\mu$ has density function
$$f(y) = \frac{1}{2\pi} \in …
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Two Different Proofs of Continuous Mapping Theorem
The theorem to prove is that if $X_n$ converges weakly to $X$, and $P(X \in D_g) = 0$ where $D_g$ is the set of discontinuity of $g$, then $g(X_n)$ converges weakly to $g(X)$.
In Durrett, this is prov …
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