Large deviations of weighted sums of independent identically distributed random variables with functionally-defined weights

I. V. Sobolev ; A. V. Shklyaev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 23 (2020) no. 1, pp. 191-206

Résumé

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 $S_n\in [x,x+\Delta_n)$.