Incremental update of Normal Distribution There's a price time series $\{p_{t}, t=1..n\}$. Is it possible to estimate Normal Distribution for every data point $N_{t}(\mu_{t}, \sigma_{t})$ efficiently (like incrementally or online calculation)?
I.e. for $N_{10} = N(p_{1},...,p_{10})$, $N_{11} = N(p_{1},...,p_{11})$, and so on.
It's possible of course to calculate it the usual way as $N_{t}=N(p_{0},..., p_{t})$ for every $t$. But it has complexity $O(n^2)$ and is slow. I'm looking for a way to speed it up.
About the assumptions of independence, stationarity and normality of the distribution. I don't know if these assumptions are true. I know that it works ok if the $N_{t}$ calculated the usual way, slow brute force way. And looking for way that would produce same $N_{t}$ but faster, with more efficient computations.
P.S.
Could the Kalman Filter be used for this? Or it solves some another task?
And, what's the proper name for this approach, incremental estimation?

Assuming that we are talking about the data that can be thought of as independent and identically distributed variables, you could just use the Welford algorithm, an algorithm that simultaneously estimates mean and variance, in an online way. Mean and variance is parameters of normal distribution, so having the parameters, you know the distribution.
If you cannot assume that the distribution is the same for all the data points, but it changes over time, you need a time-series model. In many cases, this would be much more complicated, but there are also simple time-series models like exponential smoothing that natively work in online fashion. Here however we are not necessarily assuming a normal distribution for the data. If you need the normal assumption, you need a more complicated model.
The algorithms that learn incrementally are called online-algorithms.
