When does the expected value or variance of the $t$ statistic exist? The distribution of Student's $t$ statistic is known when the random variable $x$ follows a Normal distribution. Sometimes, however, we apply it to random variables drawn from other distributions. I am curious if there are known conditions, sufficient and necessary, that the expectation of the $t$ statistic, or its variance, are known to exist (i.e. be finite).
For example, in the extreme if $x$ were drawn from a Bernoulli distribution, there would be a non-zero probability the sample variance is zero, and thus $t$ is infinite or not defined, and the expectation of $t$ does not exist. So presumably "the sample variance is positive almost surely" is a necessary condition. I am looking for more conditions like this (and ones that are easier to check).


The distribution of Student's t statistic is known when the random
variable x follows a Normal distribution. Sometimes, however, we apply
it to random variables drawn from other distributions. I am curious if
there are known conditions, sufficient and necessary, that the
expectation of the t statistic, or its variance, are known to exist
(i.e. be finite).

If the observations are normally distributed, the t-statistic follows a t-distribution under the null hypothesis, but note that with many observations it amounts to the standard normal distribution. The "other distributions" you mention for observations are the fairly large category to which the Central Limit Theorem (CLT) can be applied. So the limiting distribution for t-stat is already the standard normal distribution. Therefore, it seems to me that the conditions you mention for the finiteness of the first two moments of t-stat go back to those of the CLT.

For example, in the extreme if x were drawn from a Bernoulli
distribution, there would be a non-zero probability the sample
variance is zero, and thus t is infinite or not defined, and the
expectation of t does not exist.

Something like this will almost certainly not happen if the data come from Normal, and this is true for any large sample that meets the CLT assumptions.
Finally, the CLT implies convergence of the t-stat to the standard normal distribution, but, as noted in Whuber's comment, the CLT does not imply convergence of the moments. Indeed, convergence in the distribution does not generally imply convergence in the moments. This problem is a weakness of my argument. I will not solve it now; suggestions are welcome.
I can note, however, that even if convergence in the distribution does not imply convergence of moments, it is not true that convergence in the distribution implies or even suggests the absence of convergence of moments.
The problem we are talking about here may be of little relevance in practice, and I suspect that it is.
The rule “convergence in distribution do not imply convergence on moments” is useful also as warning about the fact that some moments may not exist. However, as mentioned above, the CLT is about the convergence of the t-stat with the standard normal distribution. I have a hard time finding an intuitive and concrete case where the t-stat approximates the standard normal distribution, but the mean and variance are significantly different compared to what that distribution should imply. Moreover, even if some cases are shown, they should be pathological and the main message of my proposal can remain.
