This paper is concerned with the small time behaviour of a Lévy process $X$. In particular, we investigate thestabilitiesof the times, ${\overline{T}}_b\left(r\right)$ and $T_b^{*}\left(r\right)$, at which $X$, started with $X_0=0$, first leaves the space-time regions $\{(t,y)\in {ℝ}^2:y\le r{t}^b,t\ge 0\}$ (one-sided exit), or $\{(t,y)\in {ℝ}^2:|y|\le r{t}^b,t\ge 0\}$ (two-sided exit), $0\le blt;1$, as $r\downarrow 0$. Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in $L^p$. In many instances these are...

The authors first establish the Marcinkiewicz-Zygmund inequalities with exponent $p$ ($1\le p\le 2$) for $m$-pairwise negatively quadrant dependent ($m$-PNQD) random variables. By means of the inequalities, the authors obtain some limit theorems for arrays of rowwise $m$-PNQD random variables, which extend and improve the corresponding results in [Y. Meng and Z. Lin (2009)] and [H. S. Sung (2013)]. It is worthy to point out that the open problem of [H. S. Sung, S. Lisawadi, and A. Volodin (2008)] can be solved easily...

The structure of linearly negative quadrant dependent random variables is extended by introducing the structure of $m$-linearly negative quadrant dependent random variables ($m=1,2,\cdots$). For a sequence of $m$-linearly negative quadrant dependent random variables $\{X_n,n\ge 1\}$ and $1<p<2$ (resp. $1\le p<2$), conditions are provided under which $n^{-1/p}{\sum }_{k=1}^n(X_k-E{X}_k)\to 0$ in $L^1$ (resp. in $L^p$). Moreover, for $1\le p<2$, conditions are provided under which $n^{-1/p}{\sum }_{k=1}^n(X_k-E{X}_k)$ converges completely to $0$. The current work extends some results of Pyke and Root (1968) and it extends and improves some...

The problem of finding simple additional conditions, for a weakly convergent sequence in $L_1$, which would suffice to imply strong convergence has been widely studied in recent years. In this Note we study this problem for Banach valued random vectors, by replacing weak convergence with a less restrictive assumption. Moreover, all the additional conditions we consider are also necessary for strong convergence, and they depend only on marginal distributions.