An additive white Gaussian noise energy-harvesting channel with an infinite-sized battery is considered. The energy arrival process is modeled as a sequence of independent and identically distributed random variables. The channel capacity $\frac{1}{2}\log(1+P)$ is achievable by the so-called best-effort and save-and-transmit schemes where $P$ denotes the battery recharge rate. This paper analyzes the save-and-transmit scheme whose transmit power is strictly less than $P$ and the best-effort scheme as a special case of save-and-transmit without a saving phase. In the finite blocklength regime, we obtain new non-asymptotic achievable rates for these schemes that approach the capacity with gaps vanishing at rates proportional to $1/\sqrt{n}$ and $\sqrt{(\log n)/n}$ respectively where~$n$ denotes the blocklength. The proof technique involves analyzing the escape probability of a Markov process. When $P$ is sufficiently large, we show that allowing the transmit power to back off from $P$ can improve the performance for save-and-transmit. The results are extended to a block energy arrival model where the length of each energy block $L$ grows sublinearly in $n$. We show that the save-and-transmit and best-effort schemes achieve coding rates that approach the capacity with gaps vanishing at rates proportional to $\sqrt{L/n}$ and $\sqrt{\max\{\log n, L\}/n}$ respectively.

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