25
votes
Accepted
"Almost all objects have property P" vs. "It is easy to test whether an object has property P"
They are separate (assuming $P \ne NP$). Consider the following property $P(x)$: $x$ is a $2n$-bit string, where either the first $n$ bits are not all zeros, or the last $n$ bits are a yes-instance ...
- 11.7k
15
votes
Accepted
What is the proof of this nonstandard version of Azuma's inequality?
I can't find a reference, so I'll just sketch the proof here.
Theorem. Let $X_1, \cdots, X_n$ be real random variables. Let $a_1, \cdots, a_n, b_1, \cdots, b_n$ be constants. Suppose that, for all $i ...
- 2,803
10
votes
Accepted
What is the connection between moments of Gaussians and perfect matchings of graphs?
This fact is a corollary of a more general theorem. Let $\gamma_1,\dots, \gamma_{2n}$ be (jointly) Gaussian random variables; we don't assume that they are independent or identically distributed. Let $...
- 3,899
10
votes
Accepted
Why is differential privacy defined over the exponential function?
This answer may be disappointing, but working on a log scale really mostly just makes the formulas nicer. The definition, as written, has the following important properties:
Composition: If $A(\cdot)$...
- 18.1k
9
votes
Accepted
Graph that maximizes minimum hitting time?
It is well known that a barbell graph (two cliques of size $n/3$ connected by a path of length $n/3$) has average hitting time $\Omega(n^3)$, but I believe the same applies to minimum hitting time (...
- 50.9k
9
votes
Accepted
What are bounded-treewidth circuits good for?
We now understand that for any fixed bound $k \in \mathbb{N}$ on the treewidth, we can convert any Boolean circuit of treewidth less than $k$ to a so-called d-SDNNF circuit, in linear time and with ...
- 8,896
9
votes
Accepted
Heterogeneous Hoeffding/McDiarmid
Yes. See for example the stronger concentration for the occupancy problem in the following note:
http://sarielhp.org/teach/17/b/lec/10_martin_II.pdf
Theorems 10.3.1 and 10.3.2.
(This is also covered ...
- 9,596
8
votes
Can the "mutual independence" condition in the Lovász local lemma be weakened?
The Lopsided Lovasz Local Lemma relaxes the mutual independence condition to negative dependence. We assume we have events $A_1, \ldots, A_n$, with a lopsidependency graph $G$ defined on $[n]$ s.t. ...
- 18.1k
7
votes
Exponential Concentration Inequality for Higher-order moments of Gaussian Random Variables
See Theorem 23 in Section 9.3 of Ryan O'Donnell's book Analysis of Boolean functions. Even though the theorem there is stated for $\pm 1$ variables, it holds for Gaussians as well (see Chapter 10 of ...
- 14.3k
7
votes
Accepted
Which graph parameters are NOT concentrated on random graphs?
Many parameters of the largest connected component are not concentrated for $G(n,p)$ if $p=1/n$ and more generally if $p$ is in the critical window. Examples are the diameter and the size of the ...
- 441
7
votes
Expected Kolmogorov complexity under Kolmogorov complexity distribution
If $\alpha$ is the answer to the 1st question then $\alpha=\infty$. Namely, for any $c $ there is an $n $ such that all strings $w $ of length at least $n $ have $K (w) \ge c$. In particular the ...
- 4,475
7
votes
Accepted
Statistical distance between uniform and biased coin
Denote the random bits by $x_1,\dots, x_n$. By definition, the statistical distance between $U$ and $D$ is at least $\Pr_U\left(\sum x_i \geq t\right) - \Pr_D\left(\sum x_i \geq t\right)$ for every $t$...
- 3,899
7
votes
Accepted
Reference for the number of samples needed to distinguish two probability distributions
So here, you only have two options: either your samples come from $p$, or from $q$ (and you know explicitly both).
By definition of the statistical/total variation distance, we have
$$
d_{TV}(p,q) = \...
- 4,451
6
votes
Can the "mutual independence" condition in the Lovász local lemma be weakened?
The formulation on p.70 of the 4th edition of The Probabilistic Method by Alon and Spencer is along the lines you state.
- 2,070
6
votes
Which graph parameters are NOT concentrated on random graphs?
Failure to concentrate happens for some counting ($\#\mathsf{P}$) properties, and maybe for many of them.
A simple example is the number of spanning subgraphs ($2^m$). The number of edges of a random ...
- 50.9k
6
votes
Accepted
Statistical Distance Growth Given K Independent Copies
See the "inclusion-exclusion" Lemma 2.2 here
https://www.cs.bgu.ac.il/~karyeh/mark-conc2.pdf .
For distributions $p,q,p',q'$, we have
$$
||p\otimes q-p'\otimes q'||
\le
||p-p'||
+
||q-q'||
-
||p-p'||
...
- 10.3k
6
votes
Accepted
Isolation Lemma over finite fields
Here is an alternative (and, hopefully, simpler) proof of Neal Young’s argument. To simplify the presentation, I take the set $\def\S{\mathcal S}\S$ of all $2^n-1$ nonempty subsets of $[n]$, and a ...
- 16.9k
5
votes
An upper bound for chi-square divergence in terms of KL divergence for general alphabets
Your definition of $\chi^2$ divergence is missing a term; namely,
$$
\chi^2(P\|Q) = \int_{\mathcal{X}} dQ\left(\frac{dP}{dQ} - 1\right)^2
= \int_{\mathcal{X}} dQ\left(\frac{dP}{dQ}\right)^2 - 1
$$
(...
- 4,451
5
votes
Reverse Chernoff bound
The Generalized Littlewood-Offord Theorem isn't exactly what you want, but it gives what I think of as a "reverse Chernoff" bound by showing that the sum of random variables is unlikely to fall within ...
- 11.9k
5
votes
Accepted
Squared Hellinger distance between Binomial(n,p) and Binomial(n+1,p)
Write $$\begin{align}
h^2(B(n,p), B(n+1,p))& = 2 \left(1 - \sum_{i=0}^{n+1}\sqrt{{n \choose i}p^i(1-p)^{n-i}{n+1 \choose i} p^i (1-p)^{n+1-i}}\right)\\
&=2\left(1 - \sum_{i=0}^{n+1}\sqrt{\left(...
- 377
5
votes
Statistical distance between uniform and biased coin
A somewhat more elementary, and slightly messier proof (or at least it feels so to me).
For convenience, write $\varepsilon = \frac{\gamma}{\sqrt{n}}$, with $\gamma\in [0,1)$ by assumption.
We ...
- 4,451
5
votes
What is the connection between moments of Gaussians and perfect matchings of graphs?
Here is a different proof, adapted from the monograph The semicircle law, free random variables and entropy. Let $X_i$ be an infinite sequence of i.i.d. variables with distribution $\Pr[X_i = 1] = \Pr[...
- 14.3k
5
votes
A coupon collector type problem with changing probabilities
Lemma. In the unbounded case, the expected number of flips is at most $\sqrt t + 3/2$.
Proof. Let r.v. $F$ be the number of flips until a head.
Then the expected number of flips is
\begin{align}
E[F]...
- 9,595
5
votes
Accepted
If I know pretty well '(a,b)', I know pretty well 'a', or 'b', or 'a xor b'
There are 4 possibilities, name them e1-e4:
e1 neither match
e2 a only matches
e3 b only matches
e4 both match
Now I restate what you want to prove: Suppose:
...
- 7,595
5
votes
Accepted
Big-O bounds on the k-th largest element of iid Gaussians
This is not a complete answer by any means, but just a quick estimate on $\mathbb{E}[\sum_{i=1}^k X_{[i]}]$ that is slightly better than the trivial bound of $O(k\sqrt{\log n})$. If this is your goal, ...
- 800
5
votes
Accepted
Relationship between Random Graph Theory and TCS
There are interesting open algorithmic problems in random graphs, which might even lead to nontrivial results about complexity classes.
For the sake of an example, consider the simplest random graph ...
- 11.3k
5
votes
Isolation Lemma over finite fields
Here is a counter-example showing that at least when $\log m \ll n$ isolation is not possible: with high probability, every weight has exponentially many sets summing to it, so no particular set is &...
- 9,595
5
votes
Evaluating asymptotic probabilities of First Order Logic Formulas?
By slightly modifying Grandjean's algorithm (reference in Emil's answer), one can show that for any fixed $k\geq 1$ and for a fixed finite (relational) language $L$, the problem of determining the ...
- 938
4
votes
Approaches for Theoretical Analysis of Estimates of Probability Distributions
Disclaimer: I am biased, in that I will suggest a survey which I have written.
What you seem to be looking for can be captured under the field of distribution testing, a subfield of Property Testing ...
- 4,451
4
votes
Minimum Spanning tree on a complete "random" graph
Let's consider a general model in which $L_n(\mu)$ is the (random) length of an MST on $K_n$, where the weight of each edge is sampled independently from a probability distribution $\mu$. When $\mu$ ...
- 18.1k
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