Journal Staff

Let $\{X_n\}_{n\geq1}$ be a sequence of i.i.d. standard Gaussian random variables, let $S_n=\sum_{i=1}^nX_i$ be the Gaussian random walk, and let $T_n=\sum_{i=1}^nS_i$ be the integrated (or iterated) Gaussian random walk. In this paper we derive the following upper and lower bounds for the conditional persistence:\begin{align*}\mathbb{P}\left\{\max_{1\leq k \leq n}T_{k} \leq 0\,\,\Big|\,\,T_n=0,S_n=0\right\}&\lesssim n^{-1/2},\\\mathbb{P}\left\{\max_{1\leq k \leq 2n}T_{k} \leq 0\,\,\Big|\,\,T_{2n}=0,S_{2n}=0\right\}&\gtrsim\frac{n^{-1/2}}{\log n},\end{align*}for $n\rightarrow\infty,$ which partially proves a conjecture by Caravenna and Deuschel (2008)

Comparison for upper tail probabilities of random series

Let $\{\xi_n\}$ be a sequence of independent and identically distributed random variables. In this paper we study the comparison for two upper tail probabilities $\mathbb{P}\{\sum_{n=1}^{\infty}a_n|\xi_n|^p\geq r\}$ and $\mathbb{P}\{\sum_{n=1}^{\infty}b_n|\xi_n|^p\geq r\}$ as $r\rightarrow\infty$ with two different real series $\{a_n\}$ and $\{b_n\}.$ The first result is for Gaussian random variables $\{\xi_n\},$ and in this case these two probabilities are equivalent after suitable scaling. The second result is for more general random variables, thus a weaker form of equivalence (namely, logarithmic level) is proved.Comment: 13 page

Limit theorems with asymptotic expansions for stochastic processes

AbstractIn this paper, we consider some families of one-dimensional locally infinitely divisible Markov processes {ηtϵ}0≤t≤T with frequent small jumps. For a smooth functional F(x[0,T]) on space D[0,T], the following asymptotic expansions for expectations are proved: as ϵ→0,EϵF(ηϵ[0,T])=EF(η0[0,T])+∑i=1sϵi/2EAiF(η0[0,T])+o(ϵs/2) for some Gaussian diffusion η0 as the weak limit of ηϵ, suitable differential operators Ai, and a positive integer s depending on the smoothness of F

Feynman-Kac formula for Levy processes and semiclassical (Euclidean) momentum representation

We prove a version of the Feynman-Kac formula for Levy processes and integro-differential operators, with application to the momentum representation of suitable quantum (Euclidean) systems whose Hamiltonians involve L\'{e}vy-type potentials. Large deviation techniques are used to obtain the limiting behavior of the systems as the Planck constant approaches zero. It turns out that the limiting behavior coincides with fresh aspects of the semiclassical limit of (Euclidean) quantum mechanics. Non-trivial examples of Levy processes are considered as illustrations and precise asymptotics are given for the terms in both configuration and momentum representations

Upper tail probabilities of integrated Brownian motions

We obtain new upper tail probabilities of $m$-times integrated Brownian motions under the uniform norm and the $L^p$ norm. For the uniform norm, Talagrand's approach is used, while for the $L^p$ norm, Zolotare's approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities (large ball probabilities) for general Gaussian random variable in Banach spaces. As applications, explicit bounds are given for the largest eigenvalue of the covariance operator, and appropriate limiting behaviors of the Laplace transforms of $m$-times integrated Brownian motions are presented as well

Probabilities of hitting a convex hull

In this note, we consider the non-negative least square method with a random matrix. This problem has connections with the probability that the origin is not in the convex hull of many random points. As related problems, suitable estimates are obtained as well on the probability that a small ball does not hit the convex hull

Impact of social media-supported flipped classroom on English as a foreign language learners’ writing performance and anxiety

As flipped classroom has received much attention from researchers and educators, some scholars have investigated the effectiveness of this teaching mode in various English as a foreign language (EFL) settings; however, such an instruction mode has been under-investigated in the Chinese EFL context. Therefore, the current study examined a flipped course’s impact on Chinese EFL learners’ writing performance and anxiety utilizing a pretest-posttest non-equivalent group quasi-experimental design. First, it selected a sample of 50 Chinese EFL learners from two intact language school classes as the participants via the convenience sampling method. Then, it randomly assigned participants of these two intact classes to a control group (n = 24), taught based on the traditional method of writing instruction, and an experimental group (n = 26), instructed based on social media-supported flipped instruction. The study used two writing tasks and a writing anxiety inventory to gather the data from the participants. The descriptive and inferential statistics results showed that the experimental group—taught based on flipped writing instruction—significantly enhanced their writing performance. Moreover, they revealed that the flipped course substantially reduced participants’ writing anxiety. Implications of such findings have been elaborated for EFL research and practice