Tagged Questions

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A sequence of nonconstant i.i.d. random variables converges with probability zero

Proove: $X_{n} iid, X_{n}$ not constant a.s. $\iff P(X_{n}$ $converges)=0$ My idea for "$\Rightarrow$": $X_{n}$ not constant a.s. $\iff \forall$ c $\in \mathbb{R}$, $\varepsilon$ > 0: ...
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stats - limiting distribution

Let $0 < p < 1$, and let $X_{n}$ have p.d.f. $f_{n}(x) = ( 1 – p ) ( n + 1 ) ( 1 – x )^{n} + p n x^{n – 1}$, for 0 < x < 1, zero elsewhere. Find the limiting distribution of $X_{n}$. ...
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stats - determine limiting distribution

Let $Y_{1} < Y_{2} < ... < Y_{n}$ be the order statistics of a random sample from a distribution with pdf $f(x) = e^{-x} , 0 < x < \infty$, zero elsewhere. Determine the limiting ...
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Convergence in Probability of a Sequence of Exponential Random Variables

If $X$ is an exponential random variable with $\lambda = 3$ and $Y_n = \frac{X^n}{n}$, I am trying to prove whether or not $Y_n$ converges in probability. My original approach was the following: ...
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$P(\sum_{k=1}^n Y_k <(1-\varepsilon)\log n)=0$ if $Y_n=\min_{1\le k\le n}X_k$ where $X_n$, $n\ge 1$ are i.i.d. Unif$(0,1)$

$P(\sum_{k=1}^n Y_k <(1-\varepsilon)\log n)=0$ if $Y_n=\min_{1\le k\le n}X_k$ where $X_n$, $n\ge 1$ are i.i.d. Unif$(0,1)$. The original question was to show $\sum_{k=1}^n Y_k/\log n$ goes to 1 in ...
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Convergence of a sequence of reciprocals

Let $X_n \geq 0$ be sequence of nonnegative random variables converging a.s. and in $L^2$ to a positive constant $c > 0$: $0 \leq X_n \xrightarrow{a.s.,L^2} c >0$ What can we say about the ...
Is there a relation between Bounded convergence theorem and convergence in distribution ? more specifically, If we have g $\geq$ 0 continuous. and $X_n \to X_{\infty}$ in distribution, Can we ...
Let $A_t$ be a bounded positive-semidefinite random matrix sequence, which converges a.s. to a positive definite matrix $A$. Let $B > 0$ be a fixed positive definite matrix. Consider the random ...