Large deviation principles in boundary problems for compound renewal processes (Q311242)

Let \(\{(\tau_k,\zeta_k)\}\) be a sequence of independent random vectors, such that \(\{(\tau_k,\zeta_k): k \geq 2\}\) are identically distributed. Further, \(\tau_k > 0\) for \(k \geq 2\) and \(\tau_1 \geq 0\). We have the sums \[ T_n = \sum_{k=1}^n \tau_k\;,\qquad S_n = \sum_{k=1}^n \zeta_k\;. \] Define the renewal process \(\nu(t) = \max\{ k \geq 0: T_k < t\}\) and the compound renewal process \(Z(t) = \sum_{k=1}^{\nu(t)} \zeta_k = S_{\nu(t)}\). For the limit theorems we consider the normalised process \(z_T(t) = T^{-1} Z(t T)\). The main results consider the asymptotics of \[ {1 \over T} \ln P[z_T \in B] \qquad \text{ as }T \to \infty \] for sets \(B\) of functions of the form \[ B_g = \{f : \sup_{t \in [0,1]} (f(t) - g(t)) \geq 0\} \] or \[ B_{g h} = \{ f : g(t) \geq f(t) \geq h(t) \text{ for all }t \in [0,1]\}. \] In particular, it turns out that not always the shortest path is the most likely path in this model.