逆チャーノフ限界

テール確率が少なくともそれほど大きいことを制限する逆チャーノフ境界があります。

すなわち、$X_1,X_2,\ldots,X_n$が独立した二項確率変数であり、$\mu=\mathbb{E}[\sum_{i=1}^n X_i]$。その後、我々は証明することができ$Pr[\sum_{i=1}^n X_i\geq (1+\delta)\mu]\geq f(\mu,\delta,n)$いくつかの機能のために$f$。

Here is an explicit proof that a standard Chernoff bound is tight up to constant factors in the exponent for a particular range of the parameters. (In particular, whenever the variables are 0 or 1, and 1 with probability 1/2 or less, and $\epsilon\in(0,1/2)$, and the Chernoff upper bound is less than a constant.)

If you find a mistake, please let me know.

Lemma 1. (tightness of Chernoff bound) Let $X$ be the average of $k$ independent, 0/1 random variables (r.v.). For any $\epsilon\in(0,1/2]$ and $p\in(0,1/2]$, assuming $\epsilon^2 p k \ge 3$,

(i) If each r.v. is 1 with probability at most $p$, then

$$\displaystyle \Pr[X\le (1-\epsilon)p] ~\ge~ \exp\big({-9\epsilon^2 pk}\big).$$

(ii) If each r.v. is 1 with probability at least $p$, then

$$\displaystyle \Pr[X\ge (1+\epsilon)p] ~\ge~ \exp\big({-9\epsilon^2 pk}\big).$$

Proof. We use the following observation:

Claim 1. If $1\le \ell \le k-1$, then $\displaystyle {k \choose \ell} ~\ge~ \frac{1}{e\sqrt{2\pi\ell}} \Big(\frac{k}{\ell}\Big)^{\ell} \Big(\frac{k}{k-\ell}\Big)^{k-\ell}$

Proof of Claim 1. By Stirling's approximation, $i!=\sqrt{2\pi i}(i/e)^ie^\lambda$ where $\lambda\in[1/(12i+1),1/12i].$

Thus, $k\choose \ell$, which is $\frac{k!}{\ell! (k-\ell)!}$, is at least

$$\frac{\sqrt{2\pi k}\,(\frac{k}{e})^k} { \sqrt{2\pi \ell}\,(\frac{\ell}{e})^\ell ~~\sqrt{2\pi (k-\ell)}\,(\frac{k-\ell}{e})^{k-\ell} } \exp\Big(\frac{1}{12k+1} - \frac{1}{12\ell} - \frac{1}{12(k-\ell)}\Big)$$ $$~\ge~ \frac{1}{\sqrt{2\pi\ell}} \Big(\frac{k}{\ell}\Big)^{\ell} \Big(\frac{k}{k-\ell}\Big)^{k-\ell}e^{-1}.$$ QED

Proof of Lemma 1 Part (i). Without loss of generality assume each 0/1 random variable in the sum $X$ is 1 with probability exactly $p$. Note $\Pr[X\le (1-\epsilon)p]$ equals the sum $\sum_{i = 0}^{\lfloor(1-\epsilon)pk\rfloor} \Pr[X=i/k]$, and $\Pr[X=i/k] = {k \choose i} p^i (1-p)^{k-i}$.

Fix $\ell = \lfloor(1-2\epsilon)pk\rfloor+1$. The terms in the sum are increasing, so the terms with index $i\ge\ell$ each have value at least $\Pr[X=\ell/k]$, so their sum has total value at least $(\epsilon pk - 2) \Pr[X=\ell/k]$. To complete the proof, we show that

$$(\epsilon pk - 2) \Pr[X=\ell/k] ~\ge~ \exp({-9\epsilon^2 pk}).$$

The assumptions $\epsilon^2pk\ge 3$ and $\epsilon\le 1/2$ give $\epsilon pk \ge 6$, so the left-hand side above is at least $\frac{2}{3}\epsilon pk\, {k \choose \ell} p^\ell(1-p)^{k-\ell}$. Using Claim 1, to bound $k\choose \ell$, this is in turn at least $A\, B$ where $A = \frac{2}{3e}\epsilon p k/ \sqrt{2\pi \ell}$ and $B= \big(\frac{k}{\ell}\big)^\ell \big(\frac{k}{k-\ell}\big)^{k-\ell} p^\ell (1-p)^{k-\ell}.$

To finish we show $A\ge \exp(-\epsilon^2pk)$ and $B \ge \exp(-8\epsilon^2 pk)$.

Claim 2. $A \ge \exp({-\epsilon^2 pk})$

Proof of Claim 2. The assumptions $\epsilon^2 pk \ge 3$ and $\epsilon\le 1/2$ imply (i) $pk\ge 12$.

By definition, $\ell \le pk + 1$. By (i), $p k \ge 12$. Thus, (ii) $\ell \,\le\, 1.1 pk$.

Substituting the right-hand side of (ii) for $\ell$ in $A$ gives (iii) $A \ge \frac{2}{3e} \epsilon \sqrt{p k / 2.2\pi}$.

The assumption, $\epsilon^2 pk \ge 3$, implies $\epsilon\sqrt{ pk} \ge \sqrt 3$, which with (iii) gives (iv) $A \ge \frac{2}{3e}\sqrt{3/2.2\pi} \ge 0.1$.

From $\epsilon^2pk \ge 3$ it follows that (v) $\exp(-\epsilon^2pk) \le \exp(-3) \le 0.04$.

(iv) and (v) together give the claim. QED

Claim 3. $B\ge \exp({-8\epsilon^2 pk})$.

Proof of Claim 3. Fix $\delta$ such that $\ell=(1-\delta)pk$.
The choice of $\ell$ implies $\delta\le 2\epsilon$, so the claim will hold as long as $B \ge \exp(-2\delta^2pk)$. Taking each side of this latter inequality to the power $-1/\ell$ and simplifying, it is equivalent to

$$\frac{\ell}{p k} \Big(\frac{k-\ell}{(1-p) k}\Big)^{k/\ell-1} ~\le~ \exp\Big(\frac{2\delta^2 pk}{\ell}\Big).$$ Substituting $\ell= (1-\delta)pk$ and simplifying, it is equivalent to $$(1-\delta) \Big(1+\frac{\delta p}{1-p}\Big)^{\displaystyle \frac{1}{(1-\delta)p}-1} ~\le~ \exp\Big(\frac{2\delta^2}{1-\delta}\Big).$$ Taking the logarithm of both sides and using $\ln(1+z)\le z$ twice, it will hold as long as $$-\delta\, +\,\frac{\delta p}{1-p}\Big(\frac{1}{(1-\delta)p}-1\Big) ~\le~ \frac{2\delta^2}{1-\delta}.$$ The left-hand side above simplifies to $\delta^2/\,(1-p)(1-\delta)$, which is less than $2\delta^2/(1-\delta)$ because $p\le 1/2$. QED

Claims 2 and 3 imply $A B \ge \exp({-\epsilon^2pk})\exp({- 8\epsilon^2pk})$. This implies part (i) of the lemma.

Proof of Lemma 1 Part (ii). Without loss of generality assume each random variable is $1$ with probability exactly $p$.

Note $\Pr[X\ge (1+\epsilon)p] = \sum_{i = \lceil(1-\epsilon)pk\rceil}^n \Pr[X=i/k]$. Fix $\hat\ell = \lceil (1+2\epsilon)pk \rceil - 1$.

The last $\epsilon pk$ terms in the sum total at least $(\epsilon pk-2)\Pr[X=\hat\ell/k]$, which is at least $\exp({-9\epsilon^2 pk})$. (The proof of that is the same as for (i), except with $\ell$ replaced by $\hat\ell$ and $\delta$ replaced by $-\hat\delta$ such that $\hat\ell = (1+\hat\delta)pk$.) QED

The Berry-Esseen theorem can give tail probability lower bounds, as long as they are higher than $n^{-1/2}$.

Another tool you can use is the Paley-Zygmund inequality. It implies that for any even integer $k$, and any real-valued random variable $X$,

$$\Pr[|X| >= \frac{1}{2}(\mathbb{E}[X^k])^{1/k}] \geq \frac{\mathbb{E}[X^k]^2}{4\mathbb{E}[X^{2k}]}$$

Together with the multinomial theorem, for $X$ a sum of $n$ rademacher random variables Paley-Zygmund can get you pretty strong lower bounds. Also it works with bounded-independence random variables. For example you easily get that the sum of $n$ 4-wise independent $\pm 1$ random variables is $\Omega(\sqrt{n})$ with constant probability.

If you are indeed okay with bounding sums of Bernoulli trials (and not, say, bounded random variables), the following is pretty tight.

Slud's Inequality*. Let $\{X_i\}_{i=1}^n$ be i.i.d. draws from a Bernoulli r.v. with $\mathbb{E}(X_1) = p$, and let integer $k\leq n$ be given. If either (a) $p\leq 1/4$ and $np \leq k$, or (b) $np \leq k \leq n(1-p)$, then

$$\text{Pr}\big[\sum_i X_i \geq k\big] \geq 1 - \Phi\left(\frac{k-np}{\sqrt{np(1-p)}}\right),$$ where $\Phi$ is the cdf of a standard normal.

(Treating the argument to $\Phi$ as transforming the standard normal, this agrees exactly with what the CLT tells you; in fact, it tells us that Binomials satisfying the conditions of the theorem will dominate their corresponding Gaussians on upper tails.)

From here, you can use bounds on $\Phi$ to get something nicer. For instance, in Feller's first book, in the section on Gaussians, it is shown for every $z>0$ that

$$\frac{z}{1+z^2}\varphi(z) < 1-\Phi(z) < \frac{1}{z}\varphi(z),$$ where $\varphi$ is the density of a standard normal. There are similar bounds in the Wikipedia article for "Q-function" as well.

Other than that, and what other people have said, you can also try using the Binomial directly, perhaps with some Stirling.

(*) Some newer statements of Slud's inequality leave out some of these conditions; I've reproduced the one in Slud's paper.

The de Moivre-Laplace Theorem shows that variables like $|T\cap S_1|$, after being suitably normalised and under certain conditions, will converge in distribution to a normal distribution. That's enough if you want constant lower bounds.

For lower bounds like $n^{-c}$, you need a slightly finer tool. Here's one reference I know of (but only by accident - I've never had the opportunity to use such an inequality myself). Some explicit lower bounds on tail probabilities of binomial distributions are given as Theorem 1.5 the book Random graphs by Béla Bollobás, Cambridge, 2nd edition, where further references are given to An introduction to probability and its applications by Feller and Foundations of Probability by Rényi.

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 a small range around any particular value (including the expectation). Perhaps it will be useful.

Formally, the theorem is as follows.

Generalized Littlewood-Offord Theorem: Let $a_1, \ldots, a_n$, and $s>0$ be real numbers such that $|a_i| \ge s$ for $1 \le i \le n$ and let $X_1, \ldots, X_n$ be independent random variables that have values zero and one. For $0 < p \le \frac{1}{2}$, suppose that $p \le \Pr[X_i = 0] \le 1-p$ for all $1 \le i \le n$. Then, for any $r \in \mathcal{R}$,

$$\Pr \left[ r \le \sum_{i=1}^{n}{a_iX_i} < r+s\right] \le \frac{c_p}{\sqrt{n}}$$ Where $c_p$ is a constant depending only on $p$.

The exponent in the standard Chernoff bound as it is stated on Wikipedia is tight for 0/1-valued random variables. Let $0<p<1$ and let $X_1,X_2,\ldots$ be a sequence of independent random variables such that for each $i$, $\Pr[X_i=1]=p$ and $\Pr[X_i=0]=1-p$. Then for every $\varepsilon>0$,

$$\begin{equation} \frac{2^{-D(p+\varepsilon\| p)\cdot n}}{n+1}\leq \Pr\left[ \sum_{i=1}^n X_i \geq (p+\varepsilon)n\right]\leq 2^{-D(p+\varepsilon\| p)\cdot n}. \end{equation}$$

Here, $D(x\| y)=x \log_2(x/y)+(1-x)\log_2((1-x)/(1-y))$, which is the Kullback-Leibler divergence between Bernoulli random variables with parameters $x$ and $y$.

As mentioned, the upper bound in the inequality above is proved on Wikipedia (https://en.wikipedia.org/wiki/Chernoff_bound) under the name "Chernoff-Hoeffding Theorem, additive form". The lower bound can be proved using e.g. the "method of types". See Lemma II.2 in [1]. Also, this is covered in the classic textbook on information theory by Cover and Thomas.

[1] Imre Csiszár: The Method of Types. IEEE Transactions on Information Theory (1998). http://dx.doi.org/10.1109/18.720546