Question

Z1,Z2,…,Zn,… is a sequence of random variables that converge in distribution to another random variable Z ;

Y1,Y2,…,Yn,… is a sequence of random variables each of which takes value in the interval (0,1) , and which converges in probability to a constant c in (0,1) ;

f(x)=x(1−x)−−−−−−−√ .

Does Znf(Yn)f(c) converge in distribution? If yes, enter the limit in terms of Z , Y and c ; if no, enter DNE.

Znf(Yn)f(c)⟶d

Answer 1

Convergence in probability is stronger than convergence in distribution. In particular, for a sequence X1, X2, X3, ⋯ to converge to a random variable X, we must have that P(|Xn−X|≥ϵ) goes to 0 as n→∞, for any ϵ>0. To say that Xn converges in probability to X, we write

Xn →p X.


If you know the definition of Convergence in Probability then you will know if Znf(Yn)f(c) converges in distribution.