A sharp uniform bound for the distribution of a sum of Bernoulli random variables

arXiv:0806.2350v1 [math.PR] 13 Jun 2008
A sharp uniform bound for the distribution of a sum of Bernoulli random variables
R. Cominetti∗ J. Vaisman†
Abstract In this paper we establish a uniform bound for the distribution of a sum S n = X1 + · · · + Xn of independent non-homogeneous Bernoulli random variables with P(Xi = 1) = pi . Specifically, we prove that σ n P(S n = i) ≤ M where σ n denotes the standard deviation of S n and √ uk 2 the constant M ∼ 0.4688 is the maximum of u → 2u e−2u ∞ k =0 ( k ! ) .
∗
σ n P(S n= i) ≤ M
(1)
1
Chebyshev, Hoeffding, and Chernoff bounds— which focus on finding sharp estimates for P(|S− E(S )| ≥ t) when S is a sum of n i.i.d. random variables and t > 0. Such large deviations bounds [1, 2, 3, 4, 5, 7, 9, 10, 12, 13] are very useful in probability and statistics, providing formulas of the type P(|S− E(S )| ≥ t) ≤ f (nt2 ) with f (0) ≥ 1 and limx→∞ f (x) = 0 usually with an exponential decay. In our particular context, this implies that P(S n= i) tends to 0 whenever i stays away from the mean E(S n ), a stronger conclusion which does not follow from (1). Thus, the main addition of Theorem 1 is that it can deal with all values of i, including those which are close to the mean. This uniform bound has already proved useful for addressing two very different and unrelated questions: (a) to study the rate of convergence of Mann’s iterates for non-expansive linear operators (see [15]); and (b) to give an approximation guarantee for an algorithm in combinatorial optimization (see [6]). We hope that the bound may be useful in other settings as well. In the rest of the paper we present the proof of Theorem 1. In §2 this proof is split into a series of basic steps, each one using only elementary tools that fit together in a surprisingly sharp way to yield the announced result. In the short final section §3 we discuss a simple extension of the main result to the case of sums and differences of Bernoullis, as well as limits of such variables which includes the difference of Poisson distributions.
Aqs qt + B (qs + qt ) + C = λ(1 − 2qr ) Aqr qt + B (qr + qt ) + C = λ(1 − 2qs ) Aqr qs + B (qr + qs ) + C = λ(1 − 2qt ). 3
Pin = P(S n = i) =
A⊂{1,...,n}
j ∈A
pj ·
j ∈A (1
− p j ).
| A| =i
Usually Pin is defined only for i = 0, . . . , n. However, Pin = P(S n = i) = 0 is also meaningful outside this range and allows to write the recursive formula
−1 n−1 Pin = pn Pin . −1 + (1 − pn )Pi
2
2.1
Reduction to a sum of 2 Binomial distributions
n The expressions Pin , σ n and Ri are continuous symmetric functions of p. We n claim that Ri (p) is maximal when the pj ’s take at most two values in (0, 1). In a different context, this result was established in [14, Vaisman] but it has not been published elsewhere.
V (x, y, z ) = x(1 − x) + y (1 − y ) + z (1 − z ) + σ n−3 (q0 )2 .
n Since the maximum of Ri (·) = σ n (·)Pin (·) is attained at q , it follows that V (·)F (·) is maximal at (qr , qs , qt ) ∈ int([0, 1]3), and therefore its gradient vanishes at this point, namely F (qr ,qs ,qt ) V (qr , qs , qt )∇F (qr , qs , qt ) + √ ∇V (qr , qs , qt ) = 0, 2 V (qr ,qs ,qt ) (qr ,qs ,qt ) this gives explicitly so that setting λ = − 2F V (qr ,qs ,qt )
Proposition 2 For all p ∈ [0, 1]n we have
n n (q ) : |{qj : 0 < qj < 1}| ≤ 2}. Ri (p) ≤ sup {Ri q ∈[0,1]n
(2)
n Proof. Let Q be the set of vectors p ∈ [0, 1]n attaining the maximum of Ri (p). Since Q is compact we may find q ∈ Q minimal in the lexicographic order. We claim that |{qj : 0 < qj < 1}| ≤ 2 from which the result follows. Assume by contradiction that q has 3 different entries 0 < qr < qs < qt < 1. Denoting q0 = (qj )j =r,s,t and q ¯j = 1 − qj we have −3 −3 Pin (q) = qr qs qt Pin qr qs qt + qr q ¯s qt + qr qs q ¯t ]Pin −3 (q0 ) + [¯ −2 (q0 ) −3 + [¯ qr q ¯s qt + q ¯r qs q ¯t + qr q ¯s q ¯t ]Pin ¯r q ¯s q ¯t Pin−3(q0 ) −1 (q0 ) + q
2
Proof of Theorem 1
n We must show that for each p = (p1 , . . . , pn ) ∈ [0, 1]n we have Ri ( p) ≤ M , n n n n n where Ri (p) = σ Pi with σ = p (1 − p ) and i i=1 i
which may be rewritten as Pin (q ) = F (qr , qs , qt ) where F (x, y, z ) = Axyz + B (xy + xz + yz ) + C (x + y + z ) + D for appropriate constants A, B, C, D. Similarly σ n (q ) = V (qr , qs , qt ) with