Non-normality of limit of random variables I have encounter the following difficulty in the study of limits of random variables. Assume that $\{X_n\}_{n\geq 1}$ is a sequence of real-valued random variables such that
$$\mathbb{E}[X_n]=\lambda_n,\,\, \mathbb{V}[X_n]=\sigma_n^2,$$
where, as usual, $\mathbb{E}$ and $\mathbb{V}$ are the expectation and variance. The condition that is satisfied is that $\sigma_n=o(\lambda_n)$.
Question: can we assure that the (conveniently normalized) sequence of random variables does NOT converge to a normal distribution $N(0,1)$? 
As far as my intuition says a simple argument (Markov, Chebyshev,...) would be enough, but I am not able to get it.
 A: From $n^2(1-p_n) \to 0$ it follows that $\mathbb{E}[\sum_{u, v \in V(G_n)} 1_{\{uv \notin E(G_n)\}}] \to 0$, 
so we have $P(\exists u, v \in V(G_n)\ \text{s.t.}\ uv \notin E(G_n)) \to 0$ by the first moment method. It is now immediate that $X_n$ will go to $\infty$ if $H$ is a finite complete graph and will converge in probability to $0$ otherwise.
A: The characteristic function framework allows you to think a ton about sequences that won't go to a normal distribution in the limit.  If I recall right, each distribution is in one to one correspondance with its characteristic function.  So look at,
$$
f_n(t) = \mathbb{E}[e^{itX_n}]
$$
and lets say for example $f_n(t)\in C^2$
$$
\log f_n(t) = \mu_n t -\frac{1}{2}\sigma_n^2 t^2 + O(\kappa_n^{(3)} t^3).
$$
Then the normalized random variable's characteristic function converges weakly to normal in the limit if and only if
$$
\lim_{n\to \infty } \log f_n(\frac{t}{\sigma_n}) - \frac{\mu_n}{\sigma_n} t = -\frac{1}{2}t^2
$$
so I would suggest playing round with functions that don't have this property until you find one you like.
