Asymptotic results for certain first-passage times and areas of renewal processes

Abstract

We consider the process { x − N ( t ) : t ≥ 0 } \{x-N(t):t\geq 0\} , where x ∈ R + x\in \mathbb {R}_+ and { N ( t ) : t ≥ 0 } \{N(t):t\geq 0\} is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of ( τ ( x ) , A ( x ) ) (\tau (x),A(x)) where τ ( x ) \tau (x) is the first-passage time of { x − N ( t ) : t ≥ 0 } \{x-N(t):t\geq 0\} to reach zero or a negative value, and A ( x ) ≔ ∫ 0 τ ( x ) ( x − N ( t ) ) d t A(x)≔\int _0^{\tau (x)}(x-N(t))dt is the corresponding first-passage (positive) area swept out by the process { x − N ( t ) : t ≥ 0 } \{x-N(t):t\geq 0\} . We remark that we can define the sequence { ( τ ( n ) , A ( n ) ) : n ≥ 1 } \{(\tau (n),A(n)):n\geq 1\} by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as x → ∞ x\to \infty in the fashion of large (and moderate) deviations.