Boundedness Index of Sequences: A Statistical Approach

Statistical boundedness weakens ordinary boundedness by allowing it to fail on a set of density zero, and this notion can be graded by an order ρ∈(0,1]. For a real or complex sequence x=(xk), we introduce the statistical boundedness index η(x), defined as the infimum of all orders ρ∈(0,1] for which x is statistically bounded of order ρ; when no such order exists, x is called index-free. For each r∈[0,1], we study the space Xr of indexed sequences with η(x)≤r, and show that these spaces increase strictly with r, forming a continuous chain that interpolates between the bounded sequences and the statistically bounded sequences. The relationship between Xr and Sbr is also clarified, and we establish the main structural and duality properties of each Xr. These results yield a fine-grained classification of the asymptotic behaviour of real and complex sequences.