By a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. Such random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in Iksanov et al. (2017) and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.

中文翻译：

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随机过程与随机时间移民的弱收敛

在随机时间具有迁移的随机过程是指具有随机响应函数（响应过程）的散粒噪声过程，其中射击在任意随机时间发生。这种随机过程概括了在 Iksanov 中引入的更新过程时期的随机过程与移民等。（2017）并且与一般分支过程中的随机特征和一般点过程生成的分支随机游走的固定生成中的计数过程非常相似。我们提供了充分的条件来确保这些过程的有限维分布对某些高斯过程的弱收敛。我们的主要结果专门针对随机时间和响应过程的几个特定实例。