Let $X_1,X_2,\dots$ be a sequence of random variables such that the distribution of their sum $S_n = \sum_{i=1}^n X_i$ satisfies a large deviation principle in the sense that there exists a lower semicontinuous function $I:\mathbb {R}\to [0,\infty ]$ such that for $a>0$ $$\liminf_{n\to\infty} \frac{1}{n}\log P(\vert S_n\vert > an) \geq -\inf_{z\in \lbrack-a,a\rbrack^C} I(z)$$ and $$\limsup_{n\to\infty} \frac{1}{n}\log P(\vert S_n\vert \geq an) \leq -\inf_{z\in (-a,a)^C} I(z).$$ Now, let $Y_1,Y_2,\dots$ be another sequence of random variables and write $R_n$ for their sum. Assume that for all natural numbers $n$ it holds that almost surely $$\vert S_n- R_n\vert \leq \log n.$$ Does the same large deviation principle or a similar one hold for the distribution of $R_n$?
One can write $P(\vert R_n\vert > an) = P\big(\vert S_n\vert > an+\mathcal{O}(\log n)\big) = P\big(\vert S_n\vert > (a+\mathcal{O}(\log n)/n)n\big)$ with $\frac{\mathcal{O}(\log n)}{n}\to 0$. But does this yield an LDP somehow with these conditions?