The range inter-event process in asymmetric birth-death random walk (Q2759392)

Let \(\{x_{t}\), \(t\geq 0\}\) be a time series defined by the birth-death process \(x_{t+ 1}=x_{t}+ \varepsilon_{t}\), \(x_0=0\), \(t\geq 1\), where \(\varepsilon_{t}\) is a random variable with \(P(\varepsilon_{t}=+ 1)=p\), \(P(\varepsilon_{t}=0)=r\), \(P(\varepsilon_{t}=-1)=q\), \(p>0\), \(q>0\), \(r\geq 0\), \(p+ q+ r=1\). This model is determined by parameters \((p,q,r)\). The range process \(\{R_{t}\), \(t\geq 0\}\) is defined as \(R_{t}=\max[x_0,x_1,\ldots,x_{t}]- \min[x_0,x_1,\ldots,x_{t}]\). Let \(\theta(a)=\inf(t\geq 0\); \(R_{t}\geq a)\). The authors study the range inter-event process \(\theta^{(k)}(a)= \theta(a+ k)-\theta(a)\). For the model with parameters \((\delta,\delta,1-2\delta)\) the authors obtain that \(E[\theta^{(k)}(a)]= k(2a+ 1+ k)/4\delta\) and NEWLINE\[NEWLINE\begin{multlined} \text{Var}[\theta^{(k)}(a)]={1\over 48\delta^2}(a+ k-1)(a+ k)(a+ k+ 1)(a+ k+ 2)+ {1\over 8\delta^2}(1-2\delta)(a+ k)(a+ k+ 1)\\ -{1\over 48\delta^2}a(a+ 1)[(a-1)(a+ 2)+ 6(1-2\delta)]. \end{multlined}NEWLINE\]NEWLINE Distribution of \(\theta^{(1)}(a)\) for the model with parameters \((\delta,\delta,1-2\delta)\) is presented.