Shocks, runs, and random sums: asymptotic behavior of the tail of the distribution function
Let \(W(k)\), \(k\geq 1\), be nonnegative random variables whose Laplace-Stieltjes transforms (LST) are defined recursively by \[ d_1(s)= a(s)P \bigl(c(s)\bigr), \quad d_{k+1}(s)= a(s)d_k(s)P \bigl(c(s) d_k(s)\bigr), \] where \(P(z)= p/(1-z+pz)\), \(0<p<1\), and \(a(s)\), \(c(s)\) are the LST of \(U(i)\), \(V(i)\), \(i\geq 1\), the \((U(i),\;V(i))\) being i.i.d. Interpretation: shocks in a system, \(U(i)\) the size of the \(i\)th shock and \(V(i)\), \(i\geq 1\), the intershock times. Let \(N(k,R)=\min \{j:U(j-i)\in R\), \(i=0,\dots,k-1\}\) and \(Z(k,R)=\sum f(U(h), V(h))\) with \(1\leq h\leq N(k,R)\). Here \(R\) is a critical region and \(Z\) measures a characteristic of the first \(k\) successive critical shocks. The authors [J. Appl. Probab. 38, No. 2, 438--448 (2001; Zbl 0987.60028)] showed that \(Z(k,R)\) has the distribution of the above \(W(k)\) with \(p=P(U(i)\in R)\) and they solved the recurrence. Here they study subexponentiality of the distribution functions \(G_k\), \(U\) and \(V\) of \(W(k)\), \(U(i)\) and \(V(i)\). By the theory of subexponentiality for subordinated distributions it is shown that if \(1-U(x)\sim c(1-H(x))\) and \(1-V(x)\sim d(1-H(x))\) with subexponential \(H\), then \(1-G_k(x)\sim c(k)(1-H(x))\), \(x\to\infty\). A similar theorem for densities is proved. The asymptotic behaviour of the remainder term in these convergences is derived.
scientific article; zbMATH DE number 1821723
scientific article