Triangular array limits for continuous time random walks (Q947153)

The authors consider a sequence of continuous time random walks indexed by a scale parameter \(c>0\) as follows. Take \(\{J_{i}^{(c)}:i\in \mathbb{N}\}\) nonnegative i.i.d. random variables representing the waiting times between particle jumps and \(T^{(c)}(n)=\sum_{i=1}^{n}J_{i}^{(c)}\) the time of the \(n\)-th jump. Let \(\{Y_{i}^{(c)}:i\in \mathbb{N}\}\) be i.i.d. random variables on \(\mathbb{R}^{d}\) representing the particle jumps and \(S^{(c)}(n)= \sum_{i=1}^{n}Y_{i}^{(c)}\) the location after \(n\) jumps. Define \(N_{t}^{(c)}=\min \{n\geq 0:T^{(c)}(n)\leq t\}\) the number of jumps by time \( t\geq 0\), and \(X_{t}^{(c)}=S^{(c)}(N_{t}^{(c)})\) the position of the particle at time \(t\geq 0\). For each \(c>0\), the random vectors \((Y_{i}^{(c)},J_{i}^{(c)})\), \(i\in \mathbb{N}\), are assumed to be i.i.d. on \( \mathbb{R}^{d}\times \mathbb{R}_{+}\). Moreover, as \(c\rightarrow \infty \), the sequence \(\{S^{(c)}(([cu]),T^{(c)}(([cu])\}_{u\geq 0}\) is assumed to converge to \(\{A(u),D(u)\}_{u\geq 0}\) in the \(J_{1}\) topology on \( D([0,\infty [ ,\mathbb{R}^{d}\times \mathbb{R}_{+})\), where \(\{A(u),D(u)\}_{u\geq 0}\) is a Lévy process on \(\mathbb{R}^{d}\times \mathbb{R}_{+}\). For \(t\geq 0\), let \(E(t)=\inf \{u\geq 0:D(u)>t\}\) and \( M(t)=A(E(t))\). The paper develops general limit theorems for \( \{X_{t}^{(c)}\}_{t\geq 0}\) as \(c\to \infty \). The main result is a density formula for the distribution of \(M(t)\). Applications to finance are also discussed.