关于带移民的随机过程收敛于平稳性的一个注记
A note on convergence to stationarity of random processes with immigration
摘要：
令$X_1, X_2,\ldots$是Skorokhod空间$D(\mathbb{R})$和$ξ_1, ξ_2, \ldots$正随机变量的随机元素，使得对$(X_1,ξ_1), (X_2,ξ_2),\ldots$是独立同分布的。随机过程$Y(t):=\sum_{k \geq 0}X_{k+1}(t-ξ_1-\ldots-ξ_k)1_{\{ξ_1+\ldots+ξ_k\leq t\}}$，$t\in\mathbb{R}$称为在更新过程的历元上有移民的随机过程.假设$ξ_1$的分布是非格的且有有限均值，而过程$X_1$衰减足够快，我们证明了在赋予$J_1$ -拓扑的$D(\mathbb{R})$上，$(Y(u+t))_{u\in\mathbb{R}}$作为$t\to\infty$的弱收敛性.本文延续了Iksanov，Marynych和Meiners（2015+）的研究路线。
英文摘要：
Let $X_1, X_2,\ldots$ be random elements of the Skorokhod space $D(\mathbb{R})$ and $ξ_1, ξ_2, \ldots$ positive random variables such that the pairs $(X_1,ξ_1), (X_2,ξ_2),\ldots$ are independent and identically distributed. The random process $Y(t):=\sum_{k \geq 0}X_{k+1}(t-ξ_1-\ldots-ξ_k)1_{\{ξ_1+\ldots+ξ_k\leq t\}}$, $t\in\mathbb{R}$, is called random process with immigration at the epochs of a renewal process. Assuming that the distribution of $ξ_1$ is nonlattice and has finite mean while the process $X_1$ decays sufficiently fast, we prove weak convergence of $(Y(u+t))_{u\in\mathbb{R}}$ as $t\to\infty$ on $D(\mathbb{R})$ endowed with the $J_1$-topology. The present paper continues the line of research initiated in Iksanov, Marynych and Meiners (2015+).
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