Second moment method

The second moment method is a technique used in probability theory and analysis to show that a random variable has positive probability to be positive. The method is often quantitative, in that one can often deducea lower bound on the probability that the random variable is larger than some constant times its expectation. The method involves comparing the second moment of random variables to the square of the first moment.

Example application of method

etup of problem

The Bernoulli bond percolation subgraph of a graph $G$ at parameter $p$ is a random subgraph obtained from $G$ by deleting every edge of $G$ with probability $1-p$, independently. The infinite complete binary tree $T$ is an infinite tree where one vertex (called the root) has two neighbors and every other vertex has three neighbors. The second moment method can be used to show that at every parameter $pin\; (1/2,1]$ with positive probability the connected component of the root in the percolation subgraph of $T$ is infinite.

Application of method

Let $K$ be the percolation component of the root, and let $T\_n$ be the set of vertices of $T$ that are at distance $n$ from the root. Let $X\_n$ be the number of vertices in $T\_ncap\; K$. To prove that $K$ is infinite with positive probability, it is enough to show that $limsup\_\{n\; oinfty\}1\_\{X\_n>0\}>0$ with positive probability. By the reverse Fatou lemma, it suffices to show that $inf\_\{n\; oinfty\}P(X\_n>0)>0$. The Cauchy-Schwarz inequality gives:Therefore, it is sufficient to show that:$inf\_n\; frac\{E(X\_n)^2\}\{E(X\_n^2)\}>0,,$that is, that the second moment is bounded from above by a constant times the first moment squared (and both are nonzero). In many applications of the second moment method, one is not able to calculate the moments precisely, but can nevertheless establish this inequality.

In this particular application, these moments can be calculated. For every specific $vin\; T\_n$,:$P(vin\; K)=p^n.$Since $|T\_n|=2^n$, it follows that:$E(X\_n)=2^n,p^n$which is the first moment.Now comes the second moment calculation. :$E(X\_n^2)=EBigl(sum\_\{vin\; T\_n\}\; sum\_\{uin\; T\_n\}1\_\{vin\; K\},1\_\{uin\; K\}Bigr)=sum\_\{vin\; T\_n\}sum\_\{uin\; T\_n\}\; P(v,uin\; K).$For each pair $v,uin\; T\_n$ let $w(v,u)$ denote the vertex in $T$ that is farthest away from the root and lies on the simple path in $T$ to each of the two vertices $v$ and $u$, and let $k(v,u)$ denote the distance from $w$ to the root. In order for $v,u$ to both be in $K$, it is necessary and sufficient for the three simple paths from $w(v,u)$ to $v,u$ and the root to be in $K$. Since the number of edges contained in the union of these three paths is $2,n-k(v,u)$, we obtain:$P(v,uin\; K)=\; p^\{2n-k(v,u)\}.$

The number of pairs $(v,u)$ such that $k(v,u)=s$ is equal to $2^s,2^\{n-s\},2^\{n-s-1\}=2^\{2n-s-1\}$, for $s=0,1,dots,n$. Hence,:$E(X\_n^2)=sum\_\{s=0\}^n\; 2^\{2n-s-1\}\; p^\{2n-s\}=frac\; 12,(2p)^\{n\}sum\_\{s=0\}^n(2p)^\{s\}=frac12,(2p)^n,frac\{(2p)^\{n+1\}-1\}\{2p-1\}le\; frac\; p\{2p-1\}\; ,E(X\_n)^2,$ which completes the proof.

Discussion

*The choice of the random variables $X\_n$ was rather natural in this setup. In some more difficult applications of the method, some ingenuity might be required in order to choose the random variables $X\_n$ for which the argument can be carried through.
*The Paley-Zygmund inequality is sometimes used instead of Cauchy-Schwarz and may occationally give more refined results.
*Under the (incorrect) assumption that the events $vin\; K$ and $uin\; K$ are always independent, one has $P(v,uin\; K)=P(vin\; K),P(uin\; K)$, and the second moment is equal to the first moment squared. The second moment method typically works in situations in which the corresponding events or random variables are “nearly independent".
*In this application, the random variables $X\_n$ are given as sums $X\_n=sum\_\{vin\; T\_n\}1\_\{vin\; K\}$. In other applications, the corresponding useful random variables are integrals $X\_n=int\; f\_n(t),dmu(t)$, where the functions $f\_n$ are random. In such a situation, one considers the product measure $mu\; imes\; mu$ and calculates:::where the last step is typically justified using Fubini's theorem.

References

*Citation | last1=Burdzy | first1=Krzysztof | last2=Adelman | first2=Omer | last3=Pemantle | first3=Robin | title=Sets avoided by Brownian motion | url=http://hdl.handle.net/1773/2194 | year=1998 | journal=Annals of Probability | volume=26 | issue=2 | pages=429–464
*Citation | last1=Lyons | first1=Russell | title=Random walk, capacity, and percolation on trees | year=1992 | journal=Annals of Probability | volume=20 | pages=2043–2088
*Citation | last1=Lyons | first1=Russell | last2=Peres | first2=Yuval | title=Probability on trees and networks | url=http://mypage.iu.edu/~rdlyons/prbtree/prbtree.html