I have the following question:
Suppose that I have a sequence of random variables $X_n$ such that all moments exists and are finite. I have that $E[X_n]\to a$, where $a$ is a finite number and also $\text{Var}(X)\to 0$. We may also assume that a density with respect to the Lebesgue measure exists. Does this imply that $X_n\to_w \delta_a$? weakly?
Thanks a lot, any ideas on conditions are welcomed! This is true for the Gaussian distribution, but I wondered to what extend we can transfer it. Literature is also welcomed!
Intuitively: the mean is almost constant, and the variance is smaller and smaller, hence we have are not too far away from the mean with a high probability.
More formally, we have $$\mathbb E\left[\left(X_n-a\right)^2\right]= \mathbb E\left[X_n^2\right]-2a\mathbb E\left[X_n\right]+a^2 =\operatorname{Var}\left(X_n\right)+\mathbb E\left[X_n\right]^2 +a^2-2a\mathbb E\left[X_n\right].$$ By assumption, $\operatorname{Var}\left(X_n\right)\to 0$ and $\mathbb E\left[X_n\right]^2 +a^2-2a\mathbb E\left[X_n\right]\to a^2+a^2-2a^2=0$, hence $X_n\to a$ in $\mathbb L^2$, hence in probability, hence in distribution.