Definition of $X_t$ in the context of Stochastic process and Time Series

Question

In the book An Introduction to Stochastic Modeling , Stochastic process is defined as :

A stochastic process is a family of random variable(s) , $X_t$ , where $t$ is a parameter running over a suitable index set, T.

If $T$ is not a random variable rather an index set , how can be $t$ a parameter ?

Isn't time series a stochastic process ? But there we say $t$ is an index and $X_t$ is an outcome at time point $t$ .

Then why should we telling $t$ a parameter in the definition of stochastic process ?

Answer 1

Parameters in a statistical sense are not realizations of a random variable:

A statistical parameter is a parameter that indexes a family of probability distributions. It can be regarded as a numerical characteristic of a population or a statistical model.

So $T$ will simply be some parameter space (for stochastic processes typically an interval in $\mathbb{R}$).

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Usually, a statistician's first association on reading the word "parameter" is to want to estimate and/or do inference on them, for instance

• for the mean or variance of a normal distribution we have sampled from (the parameter space is $\mathbb{R}\times\mathbb{R}_{>0}$)
• or for regression coefficients (the parameter space is $\mathbb{R}^{p+1}$ if you have $p$ regressors and an intercept).

However, it does not need to be the case that we necessarily should want to estimate or do inference. I have a hard time imagining a use case where we would like to estimate the time $t\in T$ on which observations in a stochastic process were sampled. However, you do have a somewhat similar question in dealing with mixture models, where you do think about deducing which of multiple component densities a particular observation came from (although I still have never seen anyone do inference on this - usually you just try to understand the entire mixture).

In any case, the $t\in T$ does satisfy the condition of "indexing a family of probability distributions", namely the $X_t$, and so it is a bona fide parameter. Of course each $X_t$ may have additional parameters that we do want to estimate or infer, e.g., in (G)ARCH modeling.

(Incidentally, in stochastic processes, $X_t$ usually denotes a random variable, namely the process at time $t$, rather than an outcome at time $t$ - it sounds a bit like you are conflating the two.)

Answer 2

The term parameter can different meanings in different settings. In statistics it is a usually a property of random variable which we want to estimate. In mathematics it is a simply a property of a mathematical object which is not constant. The stochastic processes literature usually follows mathematical conventions, so saying that $t$ is a parameter is perfectly fine.

The terminology is loosely related to terminology in calculus. When $t$ is a whole number we label it index, but when it is real number, i.e. continuous, we say parameter.

The beauty of stochastic process definition is that index set $T$ can be really anything. As long as finite dimensional distribution of a process satisfy conditions of Kolmogorov's theorem, $\{X_t, t\in T\}$ is an existing mathematical object. The parameter $t$ can be a natural number, a real number, a vector, or a set.