Limiting values of the variance and the moments of the dimension of a sum or intersection of random vector subspaces Original Research Article

Let W be an n-dimensional vector space over a field F; for each positive integer m, let the m-tuples (U1, …, Um) of vector subspaces of W be uniformly distributed; and consider the statistics Xm,1 ≔ dimF(∑i=1mUi) and Xm,2 ≔ dimF (∩i=1mUi). If F is finite of cardinality q, we determine lim E(Xm,1k), and lim E(Xm,2k), and hence, lim var(Xm,1) and lim var(Xm,2), for any k > 0, where the limits are taken as q → ∞ (for fixed n). Further, we determine whether these, and other related, limits are attained monotonically. Analogous issues are also addressed for the case of infinite F.