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A sum of observations derived by a simple random sampling design from a population of independent random variables is studied. A procedure finding a general term of Edgeworth asymptotic expansion is presented. The Lindeberg condition of asymptotic normality, Berry-Esseen bound, Edgeworth asymptotic expansions under weakened conditions and Cramer type large deviation results are derived.
Several representations of the exact cdf of the sum of squares of n independent gamma-distributed random variables Xi are given, in particular by a series of gamma distribution functions. Using a characterization of the gamma distribution by Laha, an
Let $X,X_1,dots, X_n$ be i.i.d. Gaussian random variables in a separable Hilbert space ${mathbb H}$ with zero mean and covariance operator $Sigma={mathbb E}(Xotimes X),$ and let $hat Sigma:=n^{-1}sum_{j=1}^n (X_jotimes X_j)$ be the sample (empirical)
We review a finite-sampling exponential bound due to Serfling and discuss related exponential bounds for the hypergeometric distribution. We then discuss how such bounds motivate some new results for two-sample empirical processes. Our development co
Let $pi_1$ and $pi_2$ be two independent populations, where the population $pi_i$ follows a bivariate normal distribution with unknown mean vector $boldsymbol{theta}^{(i)}$ and common known variance-covariance matrix $Sigma$, $i=1,2$. The present pap
Program synthesis has emerged as a successful approach to the image parsing task. Most prior works rely on a two-step scheme involving supervised pretraining of a Seq2Seq model with synthetic programs followed by reinforcement learning (RL) for fine-