23
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 ...
- 11.1k
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 ...
- 2,793
10
votes
Finding a biased coin using a few coin tosses
The following is a rather straight-forward $O(n \log n)$ toss algorithm.
Assume $1-\exp(-n)$ is the error probability we are aiming for. Let $N$ be some power of $2$ that is between say $100n$ and $...
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
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 ...
- 8,223
9
votes
Accepted
Upper and lower bound of binomial summation
Let me denote the sum as $S_{n,p}$. It gives the probability that a random sample from the binomial distribution $B(n,p)$ exceeds its expected value.
For $0<p<1$ constant, the central limit ...
- 15.4k
9
votes
Accepted
Approximating distributions from samples
I think it's a simple application of Hoeffding's inequality. Using your notation, let $Q_i = \frac1m C_i$, i.e. $Q$ is the empirical distribution that approximates $P$. The total variation distance ...
- 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 (...
- 50.5k
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 ...
- 9,586
8
votes
Accepted
Example of pairwise independent random process with expected max load $\sqrt{n}$
Here's how to do it.
First, choose a random $k$ between 1 and $n$ to be the "crowded bin".
Next, choose a random permutation $\pi$ of $1,2,\ldots, n-1$.
Now, for $1 \leq i \leq n-1$,
$$ \mbox{put ...
- 23.9k
8
votes
Accepted
A bounded-independence variant of the Berry-Esseen theorem
There are variants of Berry-Esseen for bounded independence, although I have not seen one which is as general as the original theorem. For example Theorem 5.1. in Diakonikolas, Kane, Nelson implies a ...
- 18.1k
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
Accepted
Number of distinct differences of $\omega(\sqrt{n})$ integers chosen from $[n]$
Assume as given that $m=\omega(\sqrt{n})$.
Fix any $\epsilon>0$. We will consider $r\in[1,n]$ with $r<(1-\epsilon)n$. The aim is to show that with high probability as $n\to\infty$, $r$ is ...
- 186
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
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 ...
- 14.2k
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
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 ...
- 4,445
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) = \...
- 4,381
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 ...
- 50.5k
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.
- 2,039
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
$$
||p\otimes q-p'\otimes q'||
\le
||p-p'||
+
||q-q'||
-
||p-p'||
...
- 10.2k
6
votes
Accepted
Isolation Lemma over finite fields
Here is an alternative (and, hopefully, simpler) proof of Neal Young’s argument. To simplify the presentation, I take the set $\def\S{\mathcal S}\S$ of all $2^n-1$ nonempty subsets of $[n]$, and a ...
- 15.4k
5
votes
Accepted
Reservoir sampling of distinct values
This is something that min-wise independent hashing is good for. (See a wikipedia explanation here.
The idea is to use a family $\mathcal{H}$ of hash functions so that when you pick a random function ...
- 131
5
votes
Accepted
Squared Hellinger distance between Binomial(n,p) and Binomial(n+1,p)
Write $$\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
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
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,
$$
\chi^2(P\|Q) = \int_{\mathcal{X}} dQ\left(\frac{dP}{dQ} - 1\right)^2
= \int_{\mathcal{X}} dQ\left(\frac{dP}{dQ}\right)^2 - 1
$$
(...
- 4,381
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 ...
- 4,381
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[...
- 14.2k
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]...
- 9,495
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