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I'm reading Introduction to Probability, 2nd Edition by Tsitsiklis and Bertsekas.

On page 218, they write "Thus, if X and Y are independent, they are also uncorrelated."

Is it possible that there could be two variables where the value of one tends to increase as the value of the other increases, even if they come from totally unrelated and disconnected processes?

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    $\begingroup$ It happens very frequently. I'd say 'unexpectedly' rather than 'by chance. Generally, one should plot pairs of predictor variables against each other in order to be aware of unanticipated correlations. Especially in economics, there can be hidden connections among variables that seem very different in type and place. $\endgroup$ – BruceET Dec 28 '18 at 9:54
  • $\begingroup$ Relevant: stats.stackexchange.com/questions/221367/… $\endgroup$ – kjetil b halvorsen Dec 28 '18 at 10:31
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    $\begingroup$ This question is confusing all the respondents because it uses two different meanings of "variable." In the quotation, $X$ and $Y$ refer to random variables, whereas your meaning of "two variables" refers to data. According to the very definition of independence, it is impossible for there to be any kind of association between $X$ and $Y$ and there is no question of "chance" operating, because $X$ and $Y$ are completely determined mathematical objects. $\endgroup$ – whuber♦ Dec 28 '18 at 15:03
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The question is confusing, and so maybe misinterpreted by some commenters and answerers. In your citation, there is two random variables $X$ and $Y$ which are independent. Then, there is a theorem saying that they are uncorrelated. It also have an easy proof, which you can find in many probability texts.

But this do not mean that if you have a sample $(X_1,Y_1), \dotsc, (X_n,Y_n)$ from $(X,Y)$, that the sample correlation coefficient will be zero! which is what the answer by @Nutle explains. But, if $n$ is large, the sampling distribution of that correlation coefficient will be concentrated close to zero.

So, yes, samples from two independent variables can seem to be correlated, by chance. Especially if $n$ is small. That just means that you risk having a type I error.

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It's a matter of change.

Yes, it's always possible that on timepoint x, two independent variables can increase or decrease at the same time. At timepoint x it looks like these variables are dependent.

However, if the data is independent they follow their own trends in most of the timepoints.

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Yes, and to add to all previous answers and comments, you can easily simulate the accidence.

I.e., this simple R example:

require(magrittr)
lapply(1:1000, function(b){
  set.seed(b)
  cor(rnorm(100), rnorm(100))
}) %>% do.call(c,.) %>% abs %>% max

Out of 1000 random draws there was a case when two independent normal random variables had a (weak) correlation of $\pm 0.34$.

This tends to decrease, of course, to zero, with increasing the size of the sample.

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