All Questions
Tagged with measure-concentration lower-bounds
9 questions
2
votes
1
answer
386
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A maximal inequality
Let $\{X_i\}_{i\in\mathbb{N}}$ be i.i.d. symmetric random variables, with $-1\leq X_i\leq 1$, $\mathbb{E}(X_i) =0$, $\mathbb{E}(X_i^2) = 1$. We have that:
$$
P\left(\bigcap_{k = 1}^{n}\frac{|\sum_{i = ...
- 63
2
votes
1
answer
336
views
The lower bound of bivariate normal distribution
Suppose $(Z_1, Z_2)$ is the zero-mean bivariate normal distribution with covariance $\left( \begin{matrix} 1 & \rho; \\ \rho & 1\end{matrix} \right)$ with positive $\rho > 0$. What I want ...
- 331
1
vote
0
answers
371
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Lower bound on the sum of the product of random variables
Let $X_i$ be the $i$-th element of the vector $X=(X_1, ..., X_m)$ of i.i.d. random variables.
I am looking for a lower bound for the expression
$\mathbb{P}((\sum^n_{i=1}\prod^{m_i}_{j=1}(X_j))^2 \geq ...
- 129
0
votes
1
answer
281
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Lower-bound on smallest singular-value of rectangular random matrix
Let $X$ be a random $N \times n$ matrix with iid entries from $\mathcal N(0, 1)$ and with $n/N =: \lambda(N,n) \le \lambda_0$, for some $\lambda_0 \in (0, 1)$. That is, $X$ is genuinely rectangular (...
- 6,853
3
votes
1
answer
88
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If $X \sim N(0,I_m)$, what is a necessary and sufficient condition on $u_m > 0$ such that $\lim\sup_{m\to \infty} P(\|X\|^2 \ge u_m|X_1|) = 1$
Let $m$ be a large positive integer and $X=(X_1,\ldots,X_m) \sim N(0,I_m)$. I wish to show that the squared norm of $X$ is much much bigger than the absolute value of any of the $X_j$'s. For example, ...
- 6,853
3
votes
1
answer
165
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Non-asymptotic tail bounds for $D_{\text{Hellinger}}(P\|\hat{P}_N)$
Let P be a distribution on a finite set of size $k$ and let $\hat{P}_N=(N_1/N,\ldots,N_k/N)$ be the empirical distribution (frequencies) from a samples of size $N$. Consider the Hellinger distance ...
- 6,853
1
vote
1
answer
91
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A distribution which is Wasserstein-close to a compactly supported distribution is almost compactly supported
I wonder whether this is true: If a distribution is very close (in the Wassertein sense) to another distribution with compact support, then the former must put only tiny amount of mass outside a ...
- 6,853
2
votes
2
answers
533
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Lower bound on misclassification rate of Lipschitz functions in terms of Lipschitz constant
Important note
@MateuszKwaśnicki in the comment section has raised a fundamental issue with the current statement of the problem. I'm trying to bugfix it.
Setup
I wish to show that a Lipschitz ...
- 6,853
3
votes
0
answers
268
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A generalization of coupon collector problem - $\geq1$ pick per experiment
Mix $T\geq1$ coupons numbered $1$ to $T$ with a set of $S\geq0$ number of dummy coupons with no numbers. Select $N\geq1$ coupons at each trial at random and put them back.
$N=1$ is standard coupon ...
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