# Questions tagged [slutsky-theorem]

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### convergence in distribution when parameters converges almost surely

let $X_n$ be sequence of random variables with the associated distribution $N(\mu_n,\beta_n+E)$. That is a sequence of normally distributed random variables with changing mean and variance. $\beta_n$ ...
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### When does $X_n\stackrel{d}{\rightarrow}X$ and $Y_n\stackrel{d}{\rightarrow}Y$ imply $X_n+Y_n\stackrel{d}{\rightarrow}X+Y$?

The question: $X_n\stackrel{d}{\rightarrow}X$ and $Y_n\stackrel{d}{\rightarrow}Y \stackrel{?}{\implies} X_n+Y_n\stackrel{d}{\rightarrow}X+Y$ I know that this does not hold in general; Slutsky's ...
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### Proving a remainder term converges to 0 in probability

So we have these definitions: σ̂^2_1= (1/n)∑(Xi−μ)^2 σ̂^2_2= (1/n)∑(Xi−Xbar)^2 I have shown that n^0.5(σ̂^2_2−σ^2)= n^0.5(σ̂^2_1−σ^2)- n^0.5(Xbar-μ)^2 I am trying to show that the remainder term ...
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### Slutsky's theorem applied to Bernoulli random variables

Suppose $X_n$ is a sequence of $n$ Bernoulli random variables with unknown $p$, and I try to get a confidence interval for $p$. Using central limit theorem, I've got \begin{align*} \frac{\bar X_n - p}...
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Let, $X_1, X_2, \ldots, X_n$ be i.i.d. RVs with mean $0$ variance $1$ and finite fourth order raw moment. Find the limiting distribution of $Z_{n}=\frac{\sqrt{n}(X_{1}X_{2}+X_{3}X_{4}+\cdots+X_{2n-1}... 1answer 99 views ### Distribution convergence of$n\Big(\log(n)\Big) X_{\min} /X_{\max}$I'm figuring out the asymptotic distribution of the following. $$n\Big(\log(n)\Big) X_{\min} /X_{\max}$$ where$X_1, \cdots, X_n \overset{iid}{\sim} \operatorname{Exp}(1)$I know it converges in ... 1answer 230 views ### Proving the delta method I am interested in proving the delta method, where we show that $$\sqrt{n}(g(Y_n) - g(\theta)) \overset{\text{Dist}}{\to} \text{N}(0, \sigma^2 g'(\theta)^2).$$ We use Taylor expansion where$$g(Y_n)... 2answers 267 views ### Intuition about Central limit theorem Suppose$X_{1}, X_{2}, ....$is a random sample from a probability distribution with mean$\mu < \infty$and$\sigma^{2} < \infty$. Let$\bar{X}_{n} = \frac{\sum X_{i}}{n}$. From the Central ... 0answers 222 views ### Proving convergence to normal distribution using the delta method The following is a question and instructor provided solution from a recent exam I took. Question: Prove that$\sqrt{n}T_n$converges in distribution to N(0, 4).$T_n$=$(\bar{X}^2 - \hat{\sigma}^2)/\...
The Slutsky's theorem: Let $\{X_n\}$, $\{Y_n\}$ be two sequences of scalar/vector/matrix random elements. If $X_n$ converges in distribution to a random element $X$ and $Y_n$ converges in ...
I am confused about some details about Slutsky's theorem: Let $\{X_n\}$, $\{Y_n\}$ be two sequences of scalar/vector/matrix random elements. If $X_n$ converges in distribution to a random ...