For a static normal distribution with fixed parameters $\mu$ and $\sigma^2$, the sample mean and sample variance can be used to estimate $\mu$ and $\sigma^2$, maybe additionally using a trimmed or winsored estimator.
What is the case when both $\mu(t)$ and $\sigma^2(t)$ are a (continuous) function of the time $t$? Is there any way to update the estimates based on previous values?
An example might be market bids and asks, where several people state their buying price for an item (say, a stock, or apples). Over time, the price may move higher or lower, but it can be frequently sampled to observe even small changes.