A symbol in stochastic process

Question

Let $\{ξ_i\}_{i=1}^n$ be i.i.d. random variable with $Eξ_i = 0$, $Dξ_i = σ^2 > 0$ and let $η_n = \frac{1}{\sigma\sqrt{n}}\sum_{i=1}^{n}ξ_i$. Prove/disprove existence of $(P) \lim_{n\to\infty} η_n$.

What is the meaning of $(P)$ in the above question?

Answer 1

$(P) \lim $ stands for convergence in probability. Here we have convergence in distribution but not convergence in probability.

Answer 2

There are different types of convergence of random variables in probability theory, for example convergence in probability, convergence in distribution, convergence almost surely.

In this case it seems to be convergence in probability.