General ballot theorem: sum of independent but not identically distributed random variables?

Question

Is there ANY ballot-type result for random walk $S_n:=\sum_{i\le n}X_i$ that allows for independent but not identically distributed random variables $X_i$, up to some uniform concentration conditions or moment bounds? By ballot-type theorem, I mean at least the upper bound with the following shape: $$\mathbb{P}(\forall J\le j\le n: S_j\le B)\le C\frac{B^C}{\sqrt{n-J}},$$

with some large constant $C>0$ independent of $B$, and some cut-off $J$(can be 1; but I guess we need the number of random variables to be large to get some concentration).

The only one I saw in the literature so far is the probability result 1 by Adam Harper, that requires $X_i$ be Gaussians with comparable variances(in between $\frac{1}{20}$ and $20$, say).

However, the result, statement (A1) on P88, is proved by relating to Lemma 5.1.8 in Lawler & Limic's Random Walk: a Modern Introduction. Then for $X_i$ not identically distributed, the statement

"But, $\mathbb{P}(S_{k+1},\cdots,S_n>S_k+1)\ge \mathbb{P}(S_{k+1}>1)\mathbb{P}(X_{k+2},\cdots,S_n-S_k>0)\ge \inf_{k}\mathbb{P}(S_k>1) \mathbb{P}(S_1,\cdots,S_n>0)$"

is not always true, because now the random walk is no longer stationary. Maybe for Gaussians with comparable variances, Harper can conduct some computations to justify it (which I don't know how); nevertheless I don't find a clue how to generalize it to other cases.

Answer 1

Yes, there are some related results for independent but not-identically distributed random variables. For asymptotic bounds assuming that $\mathbb E[X_i]=0$ and letting $B_n^2=\mathbb E[S_n^2]$ we have $$\mathbb P(\tau_x>n) \sim \frac{V(x)}{B_n},\quad n\to\infty,$$ where $\tau_x:=\inf\{n\ge 1\colon x+S_n\le 0\}$. Here $S_n=X_1+\cdots+X_n$ is a random walk with independent increments $(X_i)_{i\ge 1}$ and $V(x)$ is a function. Assumptions and more details can found in

Denisov, Sakhanenko and Wachtel (2018), First-passage times for random walks with nonidentically distributed increments. https://doi.org/10.1214/17-AOP1248

In Remark 14 of this paper there are also some inequalities for $\mathbb P(\tau_x>n)$ that follows from

Arak, T. V. (1975). The distribution of the maximum of the successive sums of independent random variables. Theory Probab. Appl. 19 245–266.