Theoretical guarantees for double spending probabilities for the Nakamoto consensus under the $k$-deep confirmation rule have been extensively studied for zero/bounded network delays and fixed mining rates. In this paper, we introduce a ruin-theoretical model of double spending for Nakamoto consensus under the $k$-deep confirmation rule when the honest mining rate is allowed to be an arbitrary function of time including the block delivery periods, i.e., time periods during which mined blocks are being delivered to all other participants of the network. Time-varying mining rates are considered to capture the intrinsic characteristics of the peer to peer network delays as well as dynamic participation of miners such as the gap game and switching between different cryptocurrencies. Ruin theory is leveraged to obtain the double spend probabilities and numerical examples are presented to validate the effectiveness of the proposed analytical method.

翻译：针对零/有界网络延迟与固定挖矿速率情形下，采用$k$深度确认规则的Nakamoto共识的双花概率理论保证已得到广泛研究。本文提出一种基于破产理论的双花模型，用于分析在$k$深度确认规则下，当诚实挖矿速率可随时间任意变化（包含区块传输周期——即新挖出区块正在向网络所有其他参与者传播的时间段）时的Nakamoto共识。时变挖矿速率的设定旨在捕捉点对点网络延迟的内在特性以及矿工动态参与行为（如间隙博弈与不同加密货币间的切换）。研究运用破产理论推导双花概率，并通过数值算例验证所提分析方法的有效性。