Inequality in probability theory
In probability theory, the van den Berg–Kesten (BK) inequality or van den Berg–Kesten–Reimer (BKR) inequality states that the probability for two random events to both happen, and at the same time one can find "disjoint certificates" to show that they both happen, is at most the product of their individual probabilities. The special case for two monotone events (the notion as used in the FKG inequality) was first proved by van den Berg and Kesten in 1985, who also conjectured that the inequality holds in general, not requiring monotonicity. Reimer later proved this conjecture. The inequality is applied to probability spaces with a product structure, such as in percolation problems.
Statement.
Let formula_1 be probability spaces, each of finitely many elements. The inequality applies to spaces of the form formula_2, equipped with the product measure, so that each element formula_3 is given the probability
formula_4
For two events formula_5, their "disjoint occurrence" formula_6 is defined as the event consisting of configurations formula_7 whose memberships in formula_8 and in formula_9 can be verified on disjoint subsets of indices. Formally, formula_10 if there exist subsets formula_11 such that:
The inequality asserts that:
formula_23
for every pair of events formula_8 and formula_22
Examples.
Coin tosses.
If formula_26 corresponds to tossing a fair coin formula_27 times, then each formula_28 consists of the two possible outcomes, heads or tails, with equal probability. Consider the event formula_8 that there exists 3 consecutive heads, and the event formula_9 that there are at least 5 heads in total. Then formula_31 would be the following event: there are 3 consecutive heads, and discarding those there are another 5 heads remaining. This event has probability at most formula_32 which is to say the probability of getting formula_8 in 10 tosses, and getting formula_9 in another 10 tosses, independent of each other.
Numerically, formula_35 formula_36 and their disjoint occurrence would imply at least 8 heads, so formula_37
Percolation.
In (Bernoulli) bond percolation of a graph, the formula_38's are indexed by edges. Each edge is kept (or "open") with some probability formula_39 or otherwise removed (or "closed"), independent of other edges, and one studies questions about the connectivity of the remaining graph, for example the event formula_40 that there is a path between two vertices formula_41 and formula_42 using only open edges. For events of such form, the disjoint occurrence formula_31 is the event where there exist two open paths not sharing any edges (corresponding to the subsets formula_15 and formula_21 in the definition), such that the first one providing the connection required by formula_18 and the second for formula_22
The inequality can be used to prove a version of the exponential decay phenomenon in the subcritical regime, namely that on the integer lattice graph formula_48 for formula_49 a suitably defined critical probability, the radius of the connected component containing the origin obeys a distribution with exponentially small tails:
formula_50
for some constant formula_51 depending on formula_52 Here formula_53 consists of vertices formula_7 that satisfies formula_55
Extensions.
Multiple events.
When there are three or more events, the operator formula_56 may not be associative, because given a subset of indices formula_57 on which formula_58 can be verified, it might not be possible to split formula_57 a disjoint union formula_60 such that formula_15 witnesses formula_62 and formula_21 witnesses formula_64. For example, there exists an event formula_65 such that formula_66
Nonetheless, one can define the formula_67-ary BKR operation of events formula_68 as the set of configurations formula_7 where there are pairwise disjoint subset of indices formula_70 such that formula_71 witnesses the membership of formula_7 in formula_73 This operation satisfies:
formula_74
whence 
formula_75
by repeated use of the original BK inequality. This inequality was one factor used to analyse the winner statistics from the Florida Lottery and identify what "Mathematics Magazine" referred to as "implausibly lucky" individuals, confirmed later by enforcement investigation that law violations were involved.
Spaces of larger cardinality.
When formula_38 is allowed to be infinite, measure theoretic issues arise. For formula_77 and formula_78 the Lebesgue measure, there are measurable subsets formula_79 such that formula_31 is non-measurable (so formula_81 in the inequality is not defined), but the following theorem still holds:
If formula_82 are Lebesgue measurable, then there is some Borel set formula_83 such that:
References.
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