Weak laws of large numbers for double sums of independent random elements in Rademacher type p and stable type p Banach spaces Original Research Article

For a double array {Vmn,m≥1,n≥1}{Vmn,m≥1,n≥1} of independent random elements in a real separable stable type View the MathML sourcep(1≤p<2) Banach space XX and sequences of random positive integers {Tn,n≥1}{Tn,n≥1} and {τn,n≥1}{τn,n≥1}, the main result provides conditions for a weak law of large numbers of the form View the MathML source∑i=1Tm∑j=1τn(Vij−c(m,n,i,j))/β(m,n)→P0as max{m,n}→∞ to hold where the c(m,n,i,j)c(m,n,i,j) are suitable elements in XX and the β(m,n)β(m,n) are suitable norming constants. The conditions are shown to completely characterize stable type View the MathML sourcep(1≤p<2) Banach spaces. Illustrative examples are provided. Moreover, for a double array of independent random elements in a real separable Rademacher type View the MathML sourcep(1≤p≤2) Banach space, a weak law of large numbers is obtained for the double sums View the MathML source∑i=1m∑j=1nVij,m≥1,n≥1.