Suppose I have a variable with a known distribution, and suppose I sample that variable k times and record the minimum. If I repeat this many times, will the median of the minimum converge to a predictable value, and if so, what is that value?

My instinct tells me that the median will converge to the value for which the cumulative probability is 1/(k+1), but I have not been able to confirm that.

Note that I found a discussion of what the expected value of the maximum would be for samples from a normal distribution: Expected value of maximum of samples from normal distribution. However, that discussion doesn't say anything about the median, which I think should be simpler.