Methodology and Computing in Applied Probability

, Volume 20, Issue 3, pp 957–973

# A BKR Operation for Events Occurring for Disjoint Reasons with High Probability

• Larry Goldstein
• Yosef Rinott
Article

## Abstract

Given events A and B on a product space $$S={\prod }_{i = 1}^{n} S_{i}$$, the set $$A \Box B$$ consists of all vectors x = (x1,…,xn) ∈ S for which there exist disjoint coordinate subsets K and L of {1,…,n} such that given the coordinates xi,iK one has that xA regardless of the values of x on the remaining coordinates, and likewise that xB given the coordinates xj,jL. For a finite product of discrete spaces endowed with a product measure, the BKR inequality
$$P(A \Box B) \le P(A)P(B)$$
(1)
was conjectured by van den Berg and Kesten (J Appl Probab 22:556–569, 1985) and proved by Reimer (Combin Probab Comput 9:27–32, 2000). In Goldstein and Rinott (J Theor Probab 20:275–293, 2007) inequality Eq. 1 was extended to general product probability spaces, replacing $$A \Box B$$ by the set Open image in new window consisting of those outcomes x for which one can only assure with probability one that xA and xB based only on the revealed coordinates in K and L as above. A strengthening of the original BKR inequality Eq. 1 results, due to the fact that Open image in new window . In particular, it may be the case that $$A \Box B$$ is empty, while Open image in new window is not. We propose the further extension Open image in new window depending on probability thresholds s and t, where Open image in new window is the special case where both s and t take the value one. The outcomes Open image in new window are those for which disjoint sets of coordinates K and L exist such that given the values of x on the revealed set of coordinates K, the probability that A occurs is at least s, and given the coordinates of x in L, the probability of B is at least t. We provide simple examples that illustrate the utility of these extensions.

## Keywords

BKR inequality Percolation Box set operation

60C05 05A20

## Notes

### Acknowledgements

We are deeply indebted to an anonymous referee for a very careful reading of two versions of this paper. The penetrating comments provided enlightened us on various measurability issues and other important, subtle points. We thank Mathew Penrose for a useful discussion and for providing some relevant references.

The work of the first author was partially supported by NSA grant H98230-15-1-0250. The second author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the program Data Linkage and Anonymisation where part of the work on this paper was undertaken, supported by EPSRC grant no EP/K032208/1.

## References

1. Arratia R, Garibaldi S, Hales AW (2015) The van den Berg—Kesten—Reimer operator and inequality for infinite spaces. Bernoulli, to appear. arXiv:1508.05337 [math.PR]
2. Arratia R, Garibaldi S, Mower L, Stark PB (2015) Some people have all the luck. Math Mag 88:196–211. arXiv:1503.02902 [math.PR]
4. Cohn DL (1980) Measure theory. Birkhäuser, Boston
5. Folland G (2013) Real analysis: modern techniques and their applications. Wiley, New York
6. Goldstein L, Rinott Y (2007) Functional BKR inequalities, and their duals, with applications. J Theor Probab 20:275–293
7. Goldstein L, Rinott Y (2015) Functional van den Berg–Kesten–Reimer inequalities and their Duals, with Applications. arXiv:1508.07267 [math.PR]
8. Gupta JC, Rao BV (1999) van den Berg-Kesten inequality for the Poisson Boolean Model for continuum Percolation. Sankhya A 61:337–346
9. Last G, Penrose MD, Zuyev S (2017) On the capacity functional of the infinite cluster of a Boolean model. Ann Appl Probab 27:1678–1701. arXiv:1601.04945 [math.PR]
10. Meester R, Roy R (2008) Continuum percolation. Cambridge University Press, Cambridge
11. Penrose M (2003) Random geometric graphs (No. 5). Oxford University Press, London
12. Reimer D (2000) Proof of the Van den Berg-Kesten conjecture. Combin Probab Comput 9:27–32
13. van den Berg J, Kesten H (1985) Inequalities with applications to percolation and reliability. J Appl Probab 22:556–569
14. van den Berg J, Fiebig U (1987) On a combinatorial conjecture concerning disjoint occurrences of events. Ann Probab 15:354–374