# Stochastic Process and Time Series (this page is not ready yet)

A Time Series can be described, i.e. modeled, as a time-discrete stochastic process. What is that –  a stochastic process? A stochastic process is a sequence of random variables indexed by (usually) time: So something like $X_1, X_2, X_3,…$, where each of the $X_i$s is a random variable. For example we could toss a coin repeatedly, lets say 4 times, and note down 1 if the result is heads and 0 otherwise. If our observations were heads, tails, tails, heads  we could note down the realization of a stochastic process as $X_1=1, X_2=0, X_3=0, X_4=1$ This particular example, where the possible outcomes are only 0 and 1 is an example of a Bernoulli Process. We can denote a stochastic process as follows:

\{X_{1}=x_{1},X_{2}=x_{2},\ldots,X_{T}=x_{T}\}=\{x_{t}\}_{t=1}^{T}.

where $X_i$’s denote random variables and T is the number of observations. When we analyze a time series that is modeled as a stochastic process, we want to deduce information about the underlying properties of the probabilistic model from this observed time series [The Analysis of Time Series An Introduction, Chris Chatfield, p.34]. This is similar to what we do, when we estimate parameters of the pdf of some population after having observed some sample of it (e.g. using Maximum Likelihood Estimation).

Remember that a discrete random variable can be described by its probability mass function (pmf) or if it is a continuous r.v. by its probability density function (pdf). So, the stochastic process is determined by the pdf/pmf of each of the r.v.s it comprises or in case that the $X_i$ are independent of each other, by their joint pdf/pmf. However, observations of a time series are rarely indepenent of each other. So here we would need the joint probability distribution of all the r.v.s. Since this is difficult to obtain, usually the first few moments of the stochastic process are used. The first moment is the expectation of a sequence [1]:

\mu_{t}=\mathbb{E}[X_{t}],

Which is can be computed for a sample in the following way:

\hat{\mu}=\frac{1}{n}\sum_{t=1}^{n}x_{t}.

Notice that this is a function of time, i.e. for each t (we assume that we take samples of the process at discrete time t) we get an expected value of the time series. This is nicely explained in this video by Dennis Sun [2]:

https://www.maths.lu.se/fileadmin/maths/personal_staff/Andreas_Jakobsson/StochProc.pdf

[1] = https://bookdown.org/gary_a_napier/time_series_lecture_notes/ChapterOne.html#objectives-of-a-time-series-analysis

[2] = https://dlsun.github.io/probability/

https://bookdown.org/compfinezbook/introcompfinr/Stochastic-Processes.html

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