Estimates for the concentration functions in the Littlewood Offord problem

Transcription

1 Estimates for the concentration functions in the Littlewood Offord problem Yulia S. Eliseeva, Andrei Yu. Zaitsev Saint-Petersburg State University, Steklov Institute, St Petersburg, RUSSIA. 203, June Saint-Petersburg Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) / 2

2 Let X, X,..., X n be independent identically distributed random variables with common distribution F = L(X). In this talk we discuss the behavior of the concentration functions of the weighted sums S a = n k= a kx k with respect to the arithmetic structure of coefficients a k. Such concentration results recently became important in connection with investigations about singular values of random matrices. In this context the problem is referred to as the Littlewood Offord problem. We formulate some refinements of results of Friedland and Sodin (2007), Rudelson and Vershynin (2009), Vershynin (20) which are proved in the recent preprints of Eliseeva and Zaitsev (202), Eliseeva, Götze and Zaitsev (202) and Eliseeva (203). Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) / 2

3 The Lévy concentration function of a random variable X is defined by the equality Q(F, λ) = sup F {[x, x + λ]}, λ 0. x R Let X, X,..., X n be independent identically distributed random variables with common distribution F = L(X), a = (a,..., a n ) R n. In the sequel, let F a denote the distribution of the sum S a = n k= a kx k, and let G be the distribution of the symmetrized random variable X = X X 2. Let M(τ) = τ 2 x 2 G{dx} + G{dx} = E min { X 2 /τ 2, }, τ > 0. () x τ x >τ The symbol c will be used for absolute positive constants. We shall write A B if A cb. Also we shall write A B if A B and B A. Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) 2 / 2

4 Theorem (Eliseeva and Zaitsev (202)). Let X, X,..., X n be i.i.d. random variables, and let a = (a,..., a n ) R n. If, for some D then, for any τ > 0, ta m α, for all m Z n and t ( Q F a, τ ) D 2 a and α > 0, [ ], D, (2) 2 a a D M(τ) + exp( c α2 M(τ)). (3) Theorem is an improvement of a result of Friedland and Sodin (2007). Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) 3 / 2

5 Theorem 2 (Eliseeva and Zaitsev (202)). Let X, X,..., X n be i.i.d. random variables, α, D > 0 and γ (0, ). Assume that ta m min{γt a, α}, for all m Z n and t [0, D]. (4) Then, for any τ > 0, ( Q F a, τ ) D a Dγ M(τ) + exp( c α2 M(τ)). (5) Theorem 2 is an improvement of a result of Rudelson and Vershynin (2009). Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) 4 / 2

6 Let L >. Theorem 3 (Eliseeva, Götze and Zaitsev (202)). Let X, X,..., X n be i.i.d. random variables. Let a = (a,..., a n ) R n. Assume that D and L 2 /M(τ), ta m f L (t a ) for all m Z n and t 2 a, τ > 0 [ ], D, (6) 2 a where t/6, f L (t) = L log(t/l), for 0 < t < el, for t el. (7) Then ( Q F a, τ ) D a D M(τ). (8) Theorem 3 is an improvement of a result of Vershynin (20). Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) 5 / 2

7 Define the least common denominator D (a) as { } D (a) = inf t > 0 : dist(ta, Z n ) < f L (t a ), (9) Theorem 4 (Eliseeva, Götze and Zaitsev (202)). Let X, X,..., X n be i.i.d. random variables. Let a = (a,..., a n ) R n. Assume that L 2 > /P, where P = P( X 0) = lim τ 0 M(τ). Then there exists a τ 0 such that L 2 = /M(τ 0 ). Moreover, the bound Q ( F a, ε ) a D (a) M(ε D (a)) (0) is valid for 0 < ε ε 0 = τ 0 /D (a). Furthermore, for ε ε 0, the bound Q ( F a, ε ) εl ε 0 a D (a) () holds. Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) 6 / 2

8 If we consider the special case, where D = /2 a, then no assumptions on the arithmetic structure of the vector a are made, and Theorems 3 imply the bound Q(F a, τ a ) a a M(τ). (2) This result follows from Esséen s (968) inequality applied to the sum of non-identically distributed random variables Y k = a k X k. For a = a 2 = = a n = n /2, inequality (2) turns into the well-known particular case: Q(F n, τ) n M(τ). (3) Inequality (3) implies as well the Kolmogorov Rogozin inequality for i.i.d. random variables: Q(F n, τ) n ( Q(F, τ)). Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) 7 / 2

9 Inequality (2) can not yield bound of better order than O(n /2 ), since the right-hand side of (2) is at least n /2. The results stated above are more interesting if D is essentially larger than /2 a. In this case one can expect the estimates of smaller order than O(n /2 ). Such estimates of Q(F a, λ) are required to study the distributions of eigenvalues of random matrices. Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) 8 / 2

10 Now we formulate the multidimensional generalizations of Theorem and Theorem 2. The concentration function of a R d valued vector Y with the distribution F = L(Y ) is defined by the equality Q(F, λ) = sup x R d P(Y x + λb), λ > 0, where B = {x R n : x }. Let X, X,..., X n be i.i.d. random variables, a = (a,..., a n ) 0, where a k = (a k,..., a kd ) R d, k =,..., n. We write A d B if A c d B and B > 0. Note that d allows constants to be exponential with respect to d. Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) 9 / 2

11 Theorem 5 (Eliseeva (203)). Let X, X,..., X n be i.i.d. random variables, a = (a,..., a n ), a k = (a k,..., a kd ) R d, k =,..., n. If for some α > 0 and D > 0 n ( t, a k m k ) 2 α 2 for all m,..., m n Z, t R d such that k= then for any τ > 0 where Q (F a, d τ ) d exp ( c α 2 M(τ) ) ( + D N = n k= max t, a k /2, t D, (4) k d D M(τ) ) d det N, ak 2 a k a k2... a k a kd a k2 a k ak a k2 a kd N k, N k = (5) a kd a k akd 2 Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) 0 / 2

12 Theorem 6 (Eliseeva (203)).Let X, X,..., X n be i.i.d. random variables. Let a = (a,..., a n ), a k R d, α > 0, D > 0, γ (0, ), and ( n ( t, a k m k ) 2) /2 min{γ t a, α} for all m,..., m n Z and t D. k= Then for any τ > 0 Q (F a, d τ ) ( d D d D γ M(τ) ) d det N + exp ( c α 2 M(τ) ). (6) Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) / 2

13 References M. Rudelson, R. Vershynin, Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62 (2009) O. Friedland, S. Sodin, Bounds on the concentration function in terms of Diophantine approximation, C. R. Math. Acad. Sci. Paris. 345 (2007) R. Vershynin, Invertibility of symmetric random matrices, arxiv: (20). To appear in Random Structures and Algorithms. Yu.S. Eliseeva, A.Yu. Zaitsev, Estimates for the concentration functions of weighted sums of independent random variables, Theory Probab. Appl. 57(202), Yu.S. Eliseeva, F. Götze, A.Yu. Zaitsev, Estimates for the concentration functions in the Littlewood Offord problem, arxiv: (202). Yu.S. Eliseeva, Multivariate estimates for the concentration functions of weighted sums of independent identically distributed random variables, arxiv: Eliseeva, Zaitsev (Saint-Petersburg State University, Steklov Institute, St Petersbourg) 2 / 2