# Large deviations for sums of random variables that are relatively close

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?

The sequences $$S_n/n$$ and $$R_n/n$$ are superexponentially close, in the sense that, for any $$\varepsilon>0$$, $$\lim_{n\rightarrow\infty}\frac{1}{n}\log P(n^{-1}|S_n - R_n| > \varepsilon) = -\infty,$$ where it is agreed that $$\log 0 = -\infty$$. It follows that $$S_n/n$$ satisfies an LDP if and only if $$R_n/n$$ does, and with the same rate function. See results on superexponential approximation, e.g. Proposition 1.19 in http://staff.utia.cas.cz/swart/lecture_notes/LDP2.pdf.