Show that law of sum of sums converges weakly to law integral of brownian motion

Question

Let $(X_k)_{k=1}^\infty$ be a sequence of independent and identically distributed random variables with zero mean and unit variance. Define, for $n\geq1$, $$S_n=\sum_{k=1}^nX_k.$$ Prove that the law of $$\frac{1}{n^{3/2}}\sum_{k=1}^nS_k$$ converges weakly to that of the random variable $$\int_0^1B_tdt$$ where $B=(B_t:t\geq0)$ is a standard Brownian motion on $\mathbb{R}$.

Hint: The sums $\sum_{k=1}^nS_k/n^{3/2}$ approximate a certain continuous operation on the space $C[0,1]$.

I really don't know where to start with this question. It seems like Skorokhod embedding and also Donsker's invariance principle could perhaps be useful, but I don't see where to start with this. I also don't see what operation these sums could be approximating. Any advice would be greatly appreciated, thanks very much!

Answer 1

The spirit of the exercise seems indeed to apply Donsker's invariance principle: we know that defining $$ W_n(t)=\frac{1}{\sqrt{n}}\left(\sum_{i=1}^{\lfloor nt\rfloor}X_i+(nt-\lfloor nt\rfloor )X_{\lfloor nt\rfloor+1}\right),n\geqslant 1, t\in [0,1], $$ the sequence $(W_n(\cdot))_{n\geqslant 1}$ converges to a standard Brownian motion $B$ in the space $C[0,1]$, the space of continuous functions endowed with the uniform norm. Since the map $F\colon C[0,1]\to\mathbb R$ defined by $F(g)=\int_0^1g(t)dt$ is continuous, the continuous mapping theorem show that $F(W_n)\to F(B)$. It remains to show that $$ F(W_n)-\frac{1}{n^{3/2}}\sum_{k=1}^nS_k\to 0\mbox{ in probability}. $$ By splitting the integral over intervals $((k-1)/n,k/n]$, we can see that $$ F(W_n)-\frac{1}{n^{3/2}}\sum_{k=1}^nS_k=\frac{1}{\sqrt{n}}\sum_{k=1}^n \int_{(k-1)/n}^{k/n}(nt-\lfloor nt\rfloor )X_{\lfloor nt\rfloor+1}dt. $$

Answer 2

Let $S(t)$ for real $t \in [0,n]$ be obtained from $S_k$ by linear interpolation.Then $$n\int_0^1 S(nt) \,dt=\int_0^n S(u) \,du= \sum_{k=1}^n \frac{S_k-S_{k-1}}{2}= (\sum_{k=1}^n S_k)-\frac{S_n}{2}\,.$$ Thus $$ n^{-3/2} \sum_{k=1}^n S_k = \int_0^1 \frac{S(nt)}{n^{1/2}}\,dt +\frac{S_n}{2n^{3/2}}\,.$$ Since the last term on the RHS tends to zero a.s. by the law of large numbers, Donsker's theorem yields the required weak convergence.