On the convergence of superpositions of point processes (Q1304461)

It is considered a sequence of two-dimensional point processes \(\{N_n, n\geq 1\}\) defined by the equations \[ N_n((0,t]\times (x,x'])= \sum^{M(nt)}_{i= 1} \sum^{A_{ni}(x')}_{j= A_{ni}(x)+ 1}X_{nij},\quad t>0,\quad x< x', \] where for each \(n\geq 1\), \(i\geq 1\), \(\{X_{nij}, j= 0,\pm 1,\pm 2,\dots\}\) are i.i.d. random variables with \(P(X_{nij}= l)= p_{nil}\), \(\sum^\infty_{l=1} p_{nil}= 1\), \(\{A_{ni}, i\geq 1\}\) be a sequence of independent ordered point processes each with index set \(R^1\) and independent of the random variables \(\{X_{nij}\}\), \(A_{ni}(x)= A_{ni}((0,x])\) if \(x\geq 0\), \(=-A_{ni}((x,0])\) if \(x<0\), and, finally, \(\{M(t), t\geq 0\}\), \(M(0)= 0\), is an ordered point process independent of everything else. Under the assumption, that there exists a function \(f(t)= bt^c+ o(t^c)\), as \(t\to\infty\), where \(b, c>0\), such that for any \(t>0\) and \(x<x'\), \[ \lim_{n\to\infty} \max_{1\leq i\leq [f(nt)]} P(A_{ni}((x, x'])\geq 1)= 0,\tag{i} \] \[ \lim_{n\to \infty} \sum^{[f(nt)]}_{i= 1} P(A_{ni}((x,x'])\geq 2)= 0,\tag{ii} \] \[ \sum^{[f(nt)]}_{i= 1} p_{nil}P(A_{ni}((x,x'])= 1)= p_l\mu((0,t]\times (x,x']),\tag{iii} \] \(\forall l\geq 1\), where \(p_l\geq 0\), \(\sum^\infty_{l= 1} p_l= 1\), \(\mu\) is a non-atomic measure on \((0,\infty)\times (-\infty,\infty)\) and \(M(t)/t^c\to \lambda\) a.s., as \(t\to\infty\), it is proved that \(\{N_n, n\geq 1\}\) converges weakly to a compound Poisson point process with parameter measure \(\mu_1\) and compounding distribution \(\{p_l, l\geq 1\}\), where \[ \mu_1((0,t]\times (x,x'])= \mu((0,(\lambda/l)^{1/c}t]\times (x,x']),\quad \forall t>0,\;x<x'. \] Some extensions of this result and examples are also discussed.