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As a start, I'm sorry if I utilize some of these terms terribly, as I'm very unfamiliar with this field.

If a series yt,yt-1, ... is weakly stationary. What would this tell use about limit of var(yt) as t approaches infinity? I found something that says the limit of E[yt] becomes a constant, but all i can find is that var(yt) "converges", now I'm not sure if that is supposed to mean it becomes 0 or becomes a fixed value (which may be 0).

tinyhippo
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$\DeclareMathOperator{\var}{var}$ $\DeclareMathOperator{\E}{E}$ That a stochastic process is weakly stationary means that $$ \E[ y_t ] = m(t) = m $$ and $$ \E[ (y_t - m(t)) (y_t' - m(t')) ] = c(t, t') = c(\Delta t), $$ where $m$ is a constant and $c$ is only a function of the time difference $\Delta t = t - t'$; i.e. both the expectation and the autocovariance are invariant under time shifts. A special case of the autocovariance is the variance $$ \E[ (y_t - m(t))^2 ] = c(0) = v, $$ where $v$ is a constant. The expectation $\E$ here is taken across the realizations of the stochastic process.

I assume that by $\var(y_t)$ you mean the variance across time, $$ \var(y_t) = \frac1t \sum_{t'=1}^t \left ( y_{t'} - \frac1t \sum_{t''=1}^t y_{t''} \right ) ^ 2, $$ and then you are interested in $\lim_{t \to \infty} \var(y_t)$?

As pointed out by Did in the comments, the notation $\var$ for the variance across time is misleading, since it should be referring to what I denoted by $v$ (or $v(t)$ for a non-stationary process) above. It is also known as the empirical or sample variance, and sometimes denoted by $\hat \sigma ^2 _t$. For the sake of simplicity, I will keep using the OP's notation in this answer.

The problem is that $\var(y_t)$ is a function of a particular realization of the process, and therefore it does not have a fixed value but is a random variable with a distribution in itself. There are several definitions for convergence of random variables. I think the one that applies here is "almost sure convergence", meaning here that there is a value $l$ such that $$ \mathrm{Pr} \left ( \lim_{t \to \infty} \var(y_t) = l \right ) = 1, $$ i.e. almost every realization of $\var(y_t)$ converges to $l$.

If $\var(y_t)$ converges in this sense, the limit will be equal to the constant point-wise variance of the process, $l = v$. Whether it converges will depend on the autocovariance function $c(\Delta t)$ of the process, in particular on whether and how fast it decreases to $0$ for increasing $\Delta t$ (whether there are long-range correlations).

A. Donda
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  • The notation var$(y_t)$ to denote the empirical variance of the process $(y_s)_{1\leqslant s\leqslant t}$ is incorrect and should be replaced. – Did Apr 17 '15 at 15:55
  • @Did, well I used the notation used in the post and gave a possible interpretation as the empirical variance; since the answer has been accepted I suppose my guess was correct. I am not aware of notation that allows to distinguish between the variance of a random variable and the empirical variance of a sequence (run into this problem often though). Which notation would you propose? – A. Donda Apr 17 '15 at 15:58
  • Yes this unfortunate notation is already in the question, but the task of an answerer is also to dispel misconceptions and/or to correct bad notations, right? Empirical variances are often denoted $\hat\sigma^2_t$. – Did Apr 17 '15 at 16:01
  • Agreed, but in the papers / books / websites I encounter, I've never seen a consistent notational distinction between the two meanings of variance. As a matter of fact, often the distinction is not even made explicit but left for the reader to figure out based on the context. I'll edit the answer to point out that problem, but I will not change the notation throughout – OK? – A. Donda Apr 17 '15 at 16:04
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    This is your answer, you can do what you want. A feature of the notation var$(y_t)$ that I find positively shocking is that the object does not depend on $y_t$ only but on every $y_s$ with $s\leqslant t$. – Did Apr 17 '15 at 16:06
  • On the other hand, the hat in $\hat \sigma ^2 _t$ implies estimation, while this quantity can also simply be considered a descriptive statistic. – A. Donda Apr 17 '15 at 16:11
  • @Did, to be absolutely notationally clear, I should also have distinguished between the process $Y_t$ and the realization $y_t$... – A. Donda Apr 17 '15 at 16:16
  • No need to, simply consider everywhere that the $y_t$ are random variables. – Did Apr 17 '15 at 16:17