I have the following puzzle to consider... Suppose that Xn is a sequence of random variables where Xn has support (0, n) Let Mn be the mode of Xn. Must the limit as n approaches infinity of P(Mn) be zero?
I am leaning towards no. I think that Xn converges in probability to zero by the law of large numbers, but I don't think this law states anything specifically about the most frequently occurring value. It seems that i.i.d samples could all have different modes so they wouldn't necessarily converge to zero. Practically, I am thinking that there is no reason the sample mode would ever need to approach the population mode...is my thinking correct here?