# convergence in distribution of sum of two normals

Let $$x_n, y_n$$ be sequences of zero mean random variables, not necessarily i.i.d. Suppose that there are finite $$\sigma_1^2,\sigma_2^2$$ such that $$x_n\overset{d}{\to} N(0,\sigma_1^2),$$ and $$y_n\overset{d}{\to} N(0,\sigma_2^2).$$ Can I say that $$x_n+y_n{\to} N(0,\sigma^2)$$ for some finite $$\sigma^2$$?

I know that I can't describe $$\sigma_2$$, but the limiting distribution of $$x_n+y_n$$ is still normally distributed?

• Are $x_n$ and $y_n$ independent? May 20, 2020 at 22:12
• no. it is not needed May 20, 2020 at 23:05
• @DilipSarwate Yes. I perceived that the limiting distribution of $x_n+y_n$ need not to be Gaussian. May 21, 2020 at 2:09
Even in the special case where $$X_n \sim N(0,\sigma_{1,n}^2), Y_n \sim N(0,\sigma_{2,n}^2)$$ where $$\{\sigma_{1,n}^2\}$$ and $$\{\sigma_{2,n}^2\}$$ are sequences of positive real numbers converging to $$\sigma_1^2$$ and $$\sigma_2^2$$ respectively, and so $$X_n \overset{d}{\to}N(0,\sigma_{1}^2), Y_n \overset{d}{\to} N(0,\sigma_{2}^2)$$, it is not possible to assert that $$X_n+Y_n$$ is a normal random variable of any kind or that $$X_n+Y_n$$ converges in distribution to a normal random variable unless it is also asserted that $$X_n, Y_n$$ are jointly normal (which implies that $$X_n, Y_n$$ are also individually (marginally) normal random variables). If $$X_n$$ and $$Y_n$$ are indeed jointly normal with correlation coefficient $$\rho_n$$ where $$\lim_{n\to\infty} \rho_n = \rho$$, then $$X_n+Y_n \overset{d}{\to}N(0,\sigma_{1}^2 + \sigma_{2}^2 + 2\rho \sigma_{1}\sigma_{2})$$