论文标题
Edgeworth的扩展，用于整数有价值的附加功能，具有均匀椭圆形的马尔可夫链
Edgeworth expansions for integer valued additive functionals of uniformly elliptic Markov chains
论文作者
论文摘要
我们获得了概率的渐近扩展，$ \ mathbb {p}（s_n = k）$的部分总和的均匀限制的整数函数$ s_n = \ sum_ {n = 1}^n f_n（x_n）均匀均匀地elliptic nopliptic nopliptic nothiptic nothiptic nopliptic nopliptic nopliptic nopliptic nopliptic nopliptic nopliptic nomogeNomogeNOMOMOGENOUS MAKEROV CHANES $。这些扩展涉及多项式和三角多项式的产物，并且它们没有其他假设。作为三角多项式的显式公式的应用，我们表明，对于每一个$ r \ geq1 \，$，$，$ s_n $ obeys obeys obeys obeys of Standard Edgeworth的订单$ r $的标准Edgeworth以每一个$ m $的条件稳定的方式，以及每一个$ m $，以及每个$ \ ell $ \ ell $ S_N $ $ s_n $的条件分配$ x_ {j_1}，...，x_ {j_ \ ell} $ mod $ m $是$ o_ \ ell（σ_n^{1-r}）$接近均匀，在选择$ j_1，...，...，j_ \ ell的选择中，j_ \ ell
We obtain asymptotic expansions for probabilities $\mathbb{P}(S_N=k)$ of partial sums of uniformly bounded integer-valued functionals $S_N=\sum_{n=1}^N f_n(X_n)$ of uniformly elliptic inhomogeneous Markov chains. The expansions involve products of polynomials and trigonometric polynomials, and they hold without additional assumptions. As an application of the explicit formulas of the trigonometric polynomials, we show that for every $r\geq1\,$, $S_N$ obeys the standard Edgeworth expansions of order $r$ in a conditionally stable way if and only if for every $m$, and every $\ell$ the conditional distribution of $S_N$ given $X_{j_1},...,X_{j_\ell}$ mod $m$ is $o_\ell(σ_N^{1-r})$ close to uniform, uniformly in the choice of $j_1,...,j_\ell$, where $σ_N=\sqrt{\text{Var}(S_N)}.$
