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Consider a random sequence $\{X_k\}_{k=1}^\infty$ of independent and identically distributed Gaussian random variables with mean $\mu \neq 0$ and variance $\sigma^2 =1$. Define a random walk described by the recursion $$S_n= S_{n-1} + \frac{1}{2}\bigg\{\frac{2X_n(X_1+\dots+X_{n-1})}{n-1} - \Big[\frac{X_1 +\dots +X_{n-1}}{n-1}\Big]^2\bigg\}$$ with $S_0 = S_1 \triangleq 0$, where . Let $$\tau = \inf\{n \geq 2: S_n \leq 0\}.$$ I am trying to prove that $\tau$ is defective, i.e., $$\mathbb{P}(\tau = \infty)>0.$$

Could anyone provide any hints on how to prove or disprove such a claim? My intuition follows from the fact that $S_n \rightarrow \infty$ almost surely. This can be seen since the expected value of the second term on the right is equal to $\frac{1}{2}\big\{\mu^2 - \frac{1}{n-1}\big\}$, thus the process has a positive drift asymptotically. Thank you for your time!

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