may be needed for $\tilde{\pi}_t^N$ to reach a small neighbourhood $U$ of this parameter value, and it may even be the case that $U$ will never be reached with large probability. Consequently, a large value of $\alpha$ should usually be compensated by a large value of $N$. In practice, we find that choosing $\alpha=0.5$, so that $\sum_{t=0}^\infty h_t=\infty$, works well and, as mentioned above, this is the default value of $\alpha$ that we use in Section \[sec:num\] for the numerical examples that rely on Algorithm \[algo:pi\_tilde\_online\].

To keep its description as simple as possible Algorithm \[algo:pi\_tilde\_online\] does not compute any approximation of the sequence $(\bar{\pi}_t)_{t\geq 0}$. A natural way to approximate this sequence using the output of Algorithm \[algo:pi\_tilde\_online\] is, for all $t\geq 0$, to sample $K$ independent random variables from $(t+1)^{-1}\sum_{s=0}^t\tilde{\pi}_s^N$ and then to use the empirical distribution of this sample as an estimate of $\bar{\pi}_t$. It turns out that such an approximation of $(\bar{\pi}_t)_{t\geq 0}$ can be computed on the fly, by adding Line \[bar\_init\] of Algorithm \[algo:pi\_tilde\] just after Line \[line:for\] of Algorithm \[algo:pi\_tilde\_online\] and by adding Line \[bar\] of Algorithm \[algo:pi\_tilde\] just after Line \[line:last\] of Algorithm \[algo:pi\_tilde\_online\]. Note also that the output of Algorithm \[algo:pi\_tilde\_online\] 