Geodesic
Large deviations of weighted sums of independent identically distributed random variables with functionally-defined weights
Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 1, pp. 191-206 
I. V. Sobolev; A. V. Shklyaev. Large deviations of weighted sums of independent identically distributed random variables with functionally-defined weights. Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 1, pp. 191-206. http://geodesic.mathdoc.fr/item/FPM_2020_23_1_a10/
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     title = {Large deviations of weighted sums of independent identically distributed random variables with functionally-defined weights},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2020_23_1_a10/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $S_n=\sum\limits_{j=1}^n a_{j,n} X_{j,n}$ be a weighted sum with independent, identically distributed steps $X_{j,n}$, $j\le n$, where $a_{j,n} = f(j/n)$ for some $f\in C^2[0,1]$. Under Cramer's condition, we prove an integro-local limit theorem for $\mathbf P\bigl(S_n\in [x,x+\Delta_n)\bigr)$ as $x/n\in [m^-,m^+]$ for some $m^-$, $m^+$ and any sequence $\Delta_n$ tending to zero slowly enough. This result covers the whole scope of normal, moderate, and large deviations. For the stochastic process $Y_n(t)$, corresponding to $S_0,\ldots,S_n$, we obtain a conditional functional limit theorem concerning convergence $Y_n(t)$ to the Brownian bridge given the condition $S_n\in [x,x+\Delta_n)$.

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