Convergence of Random Variables in Exponential Distribution

Question

I have a doubt. Given that the random variable X is exponentially distributed as Exp(1). Also X_n=x/n. Then the question is whether 'X_n' converges to probability and if yes to what. Also whether the sequence of 'X_n' are independent random variables or not.I started from the definition of convergence in probability using the density function of exponential distribution. But finally could not get. If anybody can help me. Thanks in advance.

Answer 1

For any random variable $X$ the sequence $X_n=\frac X n $tends to $0$ in probability since $P(|X_n| >\epsilon)=P(|X| >n\epsilon) \to 0$. In fact this sequence tends to $0$ almost surely.

If $X_n$'s are independent then $X$ is independent of itself. It follows that the sequence $(X_n)$ is independent iff $X$ is a constant. In our case $X$ is not a constant since it is exponentially distributed .