Consider the matrix Δn = ((I(Xi + Xj > 0)))i, j=1,2., n where (Xi) are i.i.d. and their distribution is continuous and symmetric around 0. We show that the rank rn of this matrix is equal in distribution to 2 Σn−1i=1 I(ξi = 1, ξi+1 = 0) + I(ξn = 1) (where ξii.i.d. ∼ Ber(1, 1/2). As a consequence √n(rn/n−1/2) is asymptotically normal with mean zero and variance 1/4. We also show that n−1rn converges to 1/2 almost surely.
- 1-dependent sequence
- Almost sure convergence
- Almost sure representation
- Convergence in distribution
- Large dimensional random matrix