Janson inequality

In the mathematical theory of probability, Janson's inequality is a collection of related inequalities giving an exponential bound on the probability of many related events happening simultaneously by their pairwise dependence. Informally Janson's inequality involves taking a sample of many independent random binary variables, and a set of subsets of those variables and bounding the probability that the sample will contain any of those subsets by their pairwise correlation.

Statement
Let $$\Gamma$$ be our set of variables. We intend to sample these variables according to probabilities $$p = (p_i \in [0, 1]: i \in \Gamma)$$. Let $$\Gamma_p \subseteq \Gamma$$ be the random variable of the subset of $$\Gamma$$ that includes $$i \in \Gamma$$ with probability $$p_i$$. That is, independently, for every $$i \in \Gamma: \Pr[i \in \Gamma_p]= p_i$$.

Let $$S$$ be a family of subsets of $$\Gamma$$. We want to bound the probability that any $$A \in S$$ is a subset of $$\Gamma_p$$. We will bound it using the expectation of the number of $$A \in S$$ such that $$A \subseteq \Gamma_p$$, which we call $$\lambda$$, and a term from the pairwise probability of being in $$\Gamma_p$$, which we call $$\Delta$$.

For $$A \in S$$, let $$I_A$$ be the random variable that is one if $$A \subseteq \Gamma_p$$ and zero otherwise. Let $$X$$ be the random variables of the number of sets in $$S$$ that are inside $$\Gamma_p$$: $$X = \sum_{A \in S} I_A$$. Then we define the following variables:


 * $$\lambda = \operatorname E \left[\sum_{A \in S} I_A\right] = \operatorname E[X]$$


 * $$\Delta = \frac{1}{2}\sum_{A \neq B, A \cap B \neq \emptyset} \operatorname E[I_A I_B]$$


 * $$\bar{\Delta} = \lambda + 2\Delta$$

Then the Janson inequality is:


 * $$\Pr[X = 0] = \Pr[\forall A \in S: A \not \subset \Gamma_p] \leq e^{-\lambda + \Delta} $$

and


 * $$\Pr[X = 0] = \Pr[\forall A \in S: A \not \subset \Gamma_p] \leq e^{-\frac{\lambda^2}{\bar{\Delta}}} $$

Tail bound
Janson later extended this result to give a tail bound on the probability of only a few sets being subsets. Let $$0 \leq t \leq \lambda$$ give the distance from the expected number of subsets. Let $$\phi(x) = (1 + x) \ln(1 + x) - x$$. Then we have


 * $$\Pr(X \leq \lambda - t) \leq e^{-\varphi(-t/\lambda)\lambda^2/\bar{\Delta}} \leq e^{-t^2/\left(2\bar{\Delta}\right)}$$

Uses
Janson's Inequality has been used in pseudorandomness for bounds on constant-depth circuits. Research leading to these inequalities were originally motivated by estimating chromatic numbers of random graphs.