The Jarque–Bera test is commonly used in statistics and econometrics to test the hypothesis that sample elements adhere to a normal distribution with an unknown mean and variance. This paper proposes several modifications to this test, allowing for testing hypotheses that the considered sample comes from: a normal distribution with a known mean (variance unknown); a normal distribution with a known variance (mean unknown); a normal distribution with a known mean and variance. For given significance levels, $\alpha =0.05$ and $\alpha =0.01$, we compare the power of our normality test with the most well-known and popular tests using the Monte Carlo method: Kolmogorov–Smirnov (KS), Anderson–Darling (AD), Cramér–von Mises (CVM), Lilliefors (LF), and Shapiro–Wilk (SW) tests. Under the specific distributions, 1000 datasets were generated with the sample sizes $n=25,50,75,100,150,200,250,500,$ and 1000. The simulation study showed that the suggested tests often have the best power properties. Our study also has a methodological nature, providing detailed proofs accessible to undergraduate students in statistics and probability, unlike the works of Jarque and Bera.

We prove some exponential Chebyshev inequality and derive the large deviation principle and the law of large numbers for the graphons constructed from a sequence of Erdős–Rényi random graphs with weights. Also, we obtain a new version of the large deviation principle for the number of triangles included in an Erdős–Rényi graph.

Birth–death processes form a natural class where ideas and results on large deviations can be tested. We derive a large-deviation principle under an assumption that the rate of jump down (death) grows asymptotically linearly with the population size, while the rate of jump up (birth) grows sublinearly. We establish a large-deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large-deviation probabilities. The results show interesting features of dependence of the rate functional upon the parameters of the process and the forms of scaling and normalization.

We consider an infinite directed graph with vertices numbered by integers \(\ldots,-2, -1,0,1,2,\ldots\strut\), where any pair of vertices \(j< k\) is connected by an edge \((j,k)\) that is directed from \(j\) to \(k\) and has a random weight \(v_{j,k}\in [-\infty,\infty)\). Here, \(\{v_{j,k},\: j< k\}\) is a family of independent and identically distributed random variables that take either finite values (of any sign) or the value \(-\infty\). A path in the graph is a sequence of connected edges \((j_0,j_1),(j_1,j_2),\ldots,(j_{m-1},j_m)\) (where \(j_0< j_1< \ldots < j_m\)), and its weight is the sum \(\sum\limits_{s=1}^m v_{j_{s-1},j_s}\ge -\infty\) of the weights of the edges. Let \(w_{0,n}\) be the maximal weight of all paths from \(0\) to \(n\). Assuming that \({\boldsymbol{\rm{P}}}(v_{0,1}>0)>0\), that the conditional distribution of \({\boldsymbol{\rm{P}}}(v_{0,1}\in\cdot\,\,|\, v_{0,1}>0)\) is nondegenerate, and that \({\boldsymbol{\rm{E}}}\exp (Cv_{0,1})< \infty\) for some \(C={\rm{const}} >0\), we study the asymptotic behavior of random sequence \(w_{0,n}\) as \(n\to\infty\). In the domain of the normal and moderately large deviations we obtain a local limit theorem when the distribution of random variables \(v_{i,j}\) is arithmetic and an integro-local limit theorem if this distribution is non-lattice.