New answers tagged probability-limit-theorems
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Convergence and boundedness in probability for Op + op(1)
You have to prove that $a_N+b_N$ (it is better to keep indices) converges in probability to zero. Since the sum of two sequence that converge to $0$ in probability also converges in probability, it ...
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Strong consistency of kernel density estimator
First of all, you have not explained what the function $K_h(x)$ is. From a look at the book you referenced this is defined as
$$K_h(x) = \frac{1}{h} K\left( \frac{x}{h} \right).$$
Therefore, we want ...
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Gambler's ruin in the limit (only stopping rule ruin)
There is a slick way to answer this question, in the case where $p\neq 1/2$, using Doob's martingale convergence theorem. I shall prove that
if $p\le 1/2$, then the gambler certainly goes broke.
if $...
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Gambler's ruin in the limit (only stopping rule ruin)
Your conclusion that $\ \frac{1-p}{p}\ $ is the probability of the gambler's ruin is correct if $\ p>\frac{1}{2}\ $. If $\ p\le\frac{1}{2}\ $ then the probability of his ruin is $\ 1\ $.
Let $\ \...
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Accepted
$\Pr(X_n\leq x) \to \Pr(X\leq x)$ for $x\geq 0$ and $\Pr(X_n< x) \to \Pr(X\leq x)$ for $x<0$ imply convergence in distribution?
Fix an $x<0$.
$P(X_{n}\leq x)\leq P(X_{n}<x+\frac{1}{m})\xrightarrow{n\to\infty} P(X\leq x+\frac{1}{m})$ for each $m$
And hence you have $\lim\sup_{n\to\infty}P(X_{n}\leq x)\leq P(X\leq x+\frac{...
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$\Pr(X_n\leq x) \to \Pr(X\leq x)$ for $x\geq 0$ and $\Pr(X_n< x) \to \Pr(X\leq x)$ for $x<0$ imply convergence in distribution?
Take an $x<0$, then
Step I: $\Pr(X_n = x) \to 0$.
To show this note that for every $\epsilon>0$,
$$
0\leq \Pr(X_n = x) \leq \Pr(X_n <x+\epsilon) - \Pr(X_n <x-\epsilon).
$$
Taking $\limsup$,...
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Accepted
Show a $\liminf$ statement involving probabilities
Let $a_n\leq b_n,\forall n$ and assume $a_n\to a$. We know that there exists an $N$ s.t. $a_n\in(a-\varepsilon,a+\varepsilon)$ for all $n\geq N$. So for any $\varepsilon>0$, there exists an $N$ s.t ...
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Extreme Value Theory Quantile Estimation
Calculate the quantile functions for $F$ and $G$ for your entire simulated data. If you plot histograms of $F^{-1}(u)$ and $G^{-1}(u)$, $u \in [0,1]$ it should have a uniform distribution. If it does ...
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Can weak convergence imply orders of mean and variance?
No it is not true. Assume $n^{-1/2} (Y_n - n) \Rightarrow Y$ for some nondegenerate $Y$. A counterexample is to choose $X_n$ such that $\mathbb{P}(X_n=2^n) = 1/n$ and $\mathbb{P}(X_n=Y_n)=1-1/n$. Then ...
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Accepted
An example of a sequence of RVs converging to 1 a.s., but being negative with probability converging to 1.
I'll just sum up what I said in the comments into this answer.
Almost sure convergence implies convergence in probability. Hence, what you are asking for is not possible. To be specific $P(X_{n}\leq 0)...
- 12.9k
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Accepted
Switching integration and minimization for positive random variables?
No. Let $X$ be $1$ with probability $1/2$ and $0$ otherwise (i.e., a fair coin), $Y = 1-X$, and $Z = \min(X,Y) = 0$. Then $E(X) = E(Y) = 1/2$, but $E(Z) = 0$.
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Switching integration and minimization for positive random variables?
Counter example:
Let Z = min(X,Y), with sample (x, y) = (0,0) with probability 1/3, (0,3) with probability 1/3, and (3, 0) with probability 1/3.
E[X] = E[Y] = 1. Z = 0, and E[Z] = 0. Any A between 0 ...
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