Functional limit theorems for row and column exchangeable arrays (Q1907814)

An array \(\{Y_{i,j}:1\leq i\leq m, 1\leq j\leq r\}\) is called a row and column exchangeable (RCE) array if \(\{Y_{i,j}\}\overset {D}=\{Y_{p(i),q(j)}\}\), where \(p\) and \(q\) are arbitrary permutations of \(\{1,\dots,m\}\) and \(\{1,\dots,r\}\), respectively. Assume that \(\{Y^{(n)}_{i,j}:1\leq i\leq m_n, 1\leq j\leq r_n\}\) is a sequence of RCE arrays with \(\{Y_{i,j}\}\overset {D}=\{Y^{(n)}_{\sigma_n(i),\pi_n(j)}\}\). Let \(X_n\) be the process defined on \([0,1]^2\) by \(X_n(s,t)=\sum^{[r_ns]}_{j=1} \sum^{[m_nt]}_{i=1}Y^{(n)}_{i,j}\) with \(X_n(s,0)=X_n(0,t)=0\) for all \((s,t)\in [0,1]^2\). The authors show that the various one- and two-dimensional processes arise as weak limits of the partial sum processes defined by RCE arrays \(\{Y^{(n)}_{i,j}\}\).