Strange behavior of RandomVariate and PascalDistribution

Just an extended comment: Rather than a consistent shift for the values in the minimum values for t2, it appears that the distribution is completely different than for t1, t3, and t4. Here's a figure showing that:

h[x_, label_] := Histogram[Min /@ x, {1}, "PDF", PlotRange -> {{0, 30}, {0, 0.20}}, 
  PlotLabel -> Style[label, Bold, Larger]]
GraphicsGrid[{{h[t1, "t1"], h[t2, "t2"]}, {h[t3, "t3"], h[t4, "t4"]}}, ImageSize -> Large] 

In the t1 case

Random`DistributionVector[PascalDistribution[1, 1/50], 10000, ∞]

is evaluated and in the t2 case

Random`DistributionVector[PascalDistribution[1, 1/50], 10, ∞]

The definition of this function contains a Which statement. Its first test is True for 10000 and False for 10. Its second test (there are only two) is True.

For your parameters for the PascalDistribution the switch happens between 62 and 63.