I am currently learning random walk and come across a problem concerning stopping time.
The question asks to give an example that $X_1,X_2,...$ independent r.v. with mean $0$ and variance $\sigma_i^2$, and $$E(S_T^2)\not=E (\sum_{i=1}^T \sigma_i^2) $$ when $E|T|<\infty$ and $T$ is a stopping time.
I am not sure how to construct such example as so far what I learned are all iid case and in iid case the above would be an equality. I know that in iid case and $E|T|=\infty$ there could be such inequality by letting $T=\min \{n: S_n=1\}$ of a symmetric simple random walk.
Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of independent random variables such that $$\mathbb{P}(X_n = 2^n) = \mathbb{P}(X_n = -2^n)=\frac{1}{2}$$ and set $$T := \inf\{n \geq 1; X_n = 2^n\}.$$ $T$ is a stopping time (with respect to the canonical filtration) and $$\mathbb{P}(T=n) = \frac{1}{2^n}. \tag{1} $$ This shows in particular that $$\mathbb{E}T = \sum_{n=1}^{\infty} n \mathbb{P}(T=n)<\infty.$$ Moreover, since
$$\sum_{n=1}^{N-1} 2^n = 2^{N}-2, \tag{2}$$
we have
$$S_T = \sum_{n=1}^{T-1} X_n + X_T = \sum_{n=1}^{T-1} (-2^n) + 2^T = 2.$$
Hence, $\mathbb{E}(S_T^2)=4$. On the other hand,
$$\begin{align*} \mathbb{E} \left( \sum_{n=1}^T \sigma_i^2 \right) &= \mathbb{E}\left( \sum_{N=1}^{\infty} \sum_{n=1}^N 2^i 1_{\{T=N\}} \right) \\ &\stackrel{(2)}{=} \mathbb{E} \left( \sum_{N=1}^{\infty} (2^{N+1}-2) 1_{\{T=N\}} \right) \\ &\stackrel{(1)}{=} \sum_{N=1}^{\infty} (2^{N+1}-2) \frac{1}{2^N} = \infty. \end{align*}$$