Let \(\{\) X(t), \(t\leq 0\}\) be a birth-death process in a random environment \(\{\lambda_ n,\eta_ n\}\), where \(\{\lambda_ n\), \(n\in Z\}\) and \(\{\eta_ n\), \(n\in Z\}\) are two sequences of i.i.d. positive r.v.’s, and \(\{\lambda_ n\}\) and \(\{\eta_ n\}\) are independent. The author proves that the process \(\{\) X(g(n)h(n)t)/n\(\}\) converges weakly to \(\{X_*(t)\), \(t\geq 0\}\) in the \(J_ 1\)-topology in the set of all cadlag functions on R in several general cases of \(\lambda\) ’s and \(\eta\) ’s.
Notations in the above statement are explained as following:
g(n) and h(n): convergence rate functions concerning \(\lambda\) and \(\eta\) \(respectively.\)
X\({}_*(t):=S_*^{-1}(B(V_*^{-1}(t))).\)
B(t): Brownian motion independent of \(M_*(x)\) and \(S_*(x).\)
M\({}_*(x)\) and \(S_*(x):\) limit functions of \(g(n)^{- 1}\sum^{[nx]}\lambda_ k^{-1}\) and \(h(n)^{-1}\sum^{[nx]}\eta_ k\) in sense of a.e. or d, determined by the various cases, \(respectively.\)
V\({}_*(t):=\int L(t,S_*(x))dM_*(x).\)
L(t,x): local time of B(t) at point x and \(t>0\).