Lamperti's maximal branching process is revisited, with emphasis on the description of the shape of the invariant measures in both the recurrent and transient regimes. A truncated version of this chain is exhibited, preserving the monotonicity of the original Lamperti chain supported by the integers. The Brown theory of hitting times applies to the latter chain with finite state-space, including sharp strong time to stationarity. Additional information on these hitting time problems are drawn from the quasi-stationary point of view.

中文翻译：

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重新审视 John Lamperti 的最大分支过程

重新审视了 Lamperti 的最大分支过程，重点是对周期性和瞬态状态下不变测度形状的描述。展示了该链的截断版本，保留了整数支持的原始 Lamperti 链的单调性。布朗的击中时间理论适用于具有有限状态空间的后一条链，包括极强的平稳时间。关于这些击球时间问题的附加信息是从准平稳的角度得出的。