Random walk analysis of a univariate timeseries As the title describes, I want to conduct a random walk analysis of a univariate time series $Y_t$. What are the tests and steps that you guys would suggest for this purpose?
My current thinking:

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*Analyse the autocorrelation of the changes $u_t=Y_t-Y_{t-1}$ with Ljung-Box test. Here, for different $q$'s (amount of lags considered), rejection and non-rejection of the autocorrelation changes and I am not quite sure how to interpret this.


*Second, I want to conduct an Augmented Dickey-Fuller test to test for a unit root of $Y_t$. In this context, I want to do a separrate analysis for the optimal amount of lags considered in the ADF equation. In this separrate analysis I determine the amount of lags that induces whiteness in the regression residuals. For this purpose, I am using Breusch-Godfrey's test. The reason for this additional analysis is because I want to argue that if BG test suggests that further lags are needed for residual whiteness, it is not just a random walk, even though we are not able to reject the null hypothesis of a unit root because the power of the ADF is very small if the coefficient is close to 1. Here, I have the same problem of an appropriate choice of the number of lages residuals considered in the auxiliary regression of the BG test. The resulting decision about residual whiteness is influenced by it a lot.


*Third, I want to conduct a variance ratio test. A friend of mine told me that the variance ratio test is an analysis that tests the hypothesis that the elements of $u_t$ are not only serially uncorrelated but also independent. I don't really understand where this hypothesis comes from. Shouldn't uncorrelation be enough for the linear variance increase that is analysed with a variance ratio test?
Edit: Maybe I should give more background: In the context of testing market efficiency I want to test whether a certain serious fulfills the martingale property. However, the martingale property of mean independence is not testable straight forward (Not sure about that, but most literature switches to random walk testing in context of weak market efficiency). As a random walk with i.i.d innovations also implies a martingale, I switched to testing for that as a "sufficient condition" for the series to exhibit the martingale property.
Any opinions or suggestions would be great.

The most basic property of a random walk is that its increments are IID random variables, which means that they are exchangeable.  I would start there, and use some kind of "runs test" on the increments to test the null hypothesis that they are exchangeable against the alternative hypothesis that they are not.  This is preferable to an auto-correlation test in my view, because it tests for the broader property of exchangeability, rather than messing around only with the second moment.
The other main property of the random walk is that that its increments have zero mean.  This is easy to test using a standard T-test, or any other reasonable test of a stipulated mean value.  In the comments, whuber notes a third property as being a finite second moment.  I don't consider that an essential property of the concept, so I wouldn't bother testing it, but you should consider whether you regard this as an important property to test.  If you want to test for a finite second variance then you can use a "tail plot" and various corresponding tests that look at the rate of decay in the tails.
