Autocorrelation definition I am formatting a statistics proof, and I wanted to make sure that I have the definition of autocorrelation correct.
Is it the case that the autocorrelation of a continuous variable is the same as the correlation of that same variable with itself at different points in time?
 A: Short answer: Yes, that is one form of autocorrelation.

Longer answer: The prefix "auto-" derives from the Greek word autos for "self".  The concept of "auto-correlation" in statistics just refers to self-correlation.  It arises whenever you have a set of random variables $X_t$ indexed by an index $t$, where you can properly regard the values as representing the "same" variable, measured under different conditions (e.g., at different points in time, space, etc.).  In such a context, you can specify the correlation between different values in the sequence, and these correlation values are called "auto-correlation", since they refer to the correlation between the "same" quantity at different points in time/space.  The key requirement is that the random variables under consideration represent the same underlying quantity, with differences attributable to measurement under different conditions (e.g., at different points in time, space, etc.).
The notion of auto-correlation hinges on the interpretive issue of whether or not random variables in a sequence represent the "same" quantity or not.  For this reason, the concept is not solely a mathematical concept, and so there is no formal mathematical definition of what auto-correlation is.  Formally speaking, auto-correlation is just regular correlation between random variables, but in a context where those random variables can be regarded as representing the "same" underlying quantity.  This is why, when you search for a "formal definition" of auto-correlation, you will find a lot of hand-waving --- it is not solely a mathematical concept.
It is also worth noting that time-series models that invoke auto-correlation tend to do so in a context where some kind of stationarity condition holds (e.g., weak stationarity), and in this case the auto-correlation depends only on the "lag" between the time-series values.  This leads some analysts to incorrectly believe that the concept of "auto-correlation" refers exclusively to this kind of lagged correlation in ARMA models, or similar models.  Actually, while these are examples of auto-correlation, there is no requirement that auto-correlation meet the requirements of stationarity, and it is a more general concept that can refer to any correlation structure between random variables that represent the "same" underlying quantity when varied over time, space, etc.
A: There are many definition of autocorrelation.
I like these 2:-

"correlation between the elements of a series and others from the same series separated from them by a given interval."~Google
"Autocorrelation, also known as serial correlation, is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time lag between them."~Wiki

Perhaps,you would ask about Series of Continuous Random Variable........ And correlation and autocorrelation is not always same in given situation.
Now, Consider auto-correlation of lag k, if the series is stationary or at least weakly stationary, then autocorrelation of a series Continuous Random Variable with lag k is the same as all the correlation of that same variable with itself with lag k at different points in time.
if not, then not.
