# Tagged Questions

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### Sequence of connected sets converging to a disconnected set [on hold]

Let us have the disconnected set $\mathcal{X} = \{0\} \cup [\underline{x},\overline{x}]$, with $0 > \underline{x} > \overline{x}$. Is it possible to define a sequence of connected sets ...
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### Berry-Esseen result for triangular arrays

Let $\{\{X_{n1},\ldots,X_{nn}\},n=1,2,\ldots\}$ be a triangular array of independent random variables where each row contains identically distributed random variables. Let $E[X_{n1}]=\mu_n<\infty$ ...
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### Central Limit Theorem for Functions of Uniform Random Variables on the Sphere

For each $n$, let $\boldsymbol{X}_1$ and $\boldsymbol{X}_2$ be random length-$n$ sequences, distributed uniformly on the surface of the spheres of radii $nP_1$ and $nP_2$. Let $\boldsymbol{Y}$ = ...
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### Central Limit Theorem for additive function of permutations of sequences

Fix the finite sets $\mathcal{X}$ and $\mathcal{Y}$, and probability mass functions $P_X(x)$ and $P_Y(y)$ on these sets. Assume each value of $P_X(x)$ and $P_Y(y)$ is rational. For each $n$ such ...
Consider the triangular array $X_{n,k}$ such that, for each $n>0$, the variables $(X_{n,1},\cdots,X_{n,n})$ have the following properties: For any given $1 \le L \le n$, all subsets of ...
### Convergence of $L^p$ means and/or norms
Dear all, I have a question about convergence of $L^p$-means. It can be shown (Inequalities, Theorem 193, Hardy, Littlewood, Polya) that \$\forall f \in L^p(D,\mu)\cap L^\infty(D,\mu), M_p(f) = ...