On \(B_ \varphi\)-convex spaces (Q2366406)

We generalize the notion of \(B\)-convex space and establish strengthened laws of large numbers for weighted sums of random elements with respect to such a generalization. The generalization's essence is concerned with the replacement of power function in the convexity definition by an arbitrary function \(\varphi\) which satisfies certain conditions on growth. The introduced \(B_ \varphi\)-convex spaces are characterized by the laws of large numbers for weighted sums, while the previously known convex spaces were characterized by the Marcinkiewicz laws of large numbers. We also establish conditions under which the generalized space is of stable type \(p\).