Let ${\left(M_n\right)}_{n\ge 0}$ be a zero-mean martingale with canonical filtration ${\left({ℱ}_n\right)}_{n\ge 0}$ and stochastically $L_2$-bounded increments $Y_1,Y_2,...,$ which means that $P\left(\right|Y_n|>t|{ℱ}_{n-1})\le 1-H\left(t\right)$ a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let $V^2={\sum }_{n\ge 1}E\left(Y_n^2\right|{ℱ}_{n-1})$. It is the main result of this paper that each such martingale is a.s. convergent on V < ∞ and recurrent on V = ∞, i.e. $P(M_n\in [-c,c]i.o.|V=\infty )=1$ for some c > 0. This generalizes a recent result by Durrett, Kesten and Lawler [4] who consider the case of only finitely many square-integrable increment distributions....

Les distributions non paramétriques de survie trouvent, de plus en plus, des applications dans des domaines très variés, à savoir : théorie de fiabilité et analyse de survie, files d’attente, maintenance, gestion de stock, théorie de l’économie, $...$ L’objet de ce travail est d’utiliser les bornes inférieures et supérieures (en terme de la moyenne) des fonctions de fiabilité appartenant aux classes de distribution de type $IFR,DFR,NBU$ et $NWU$, présentées par Sengupta (1994), pour l’évaluation de certaines caractéristiques....

Les distributions non paramétriques de survie trouvent, de plus en plus, des applications dans des domaines très variés, à savoir: théorie de fiabilité et analyse de survie, files d'attente, maintenance, gestion de stock, théorie de l'économie, ... L'objet de ce travail est d'utiliser les bornes inférieures et supérieures (en terme de la moyenne) des fonctions de fiabilité appartenant aux classes de distribution de type IFR, DFR, NBU et NWU, présentées par Sengupta (1994), pour l'évaluation de...

The paper deals with several questions of the diffusion approximation. The goal of this paper is to create the general method of reducting the dimension of the model with the aid of the diffusion approximation. Especially, two dimensional random variables are approximated by one-dimensional diffusion process by replacing one of its coordinates by a certain characteristic, e.g. by its stationary expectation. The suggested method is used for several different systems. For instance, the method is applicable...

Let $(X_t,t\ge 0)$ be a Lévy process started at $0$, with Lévy measure $\nu$. We consider the first passage time $T_x$ of $(X_t,t\ge 0)$ to level $x\&gt;0$, and $K_x:=X_{T_x}-𝑥$ the overshoot and $L_x:=x-X_{T_{{𝑥}^{-}}}$ the undershoot. We first prove that the Laplace transform of the random triple $(T_x,K_x,L_x)$ satisfies some kind of integral equation. Second, assuming that $\nu$ admits exponential moments, we show that $(\widetilde{T_x},K_x,L_x)$ converges in distribution as $x\to \infty$, where $\widetilde{T_x}$ denotes a suitable renormalization of $T_x$.

Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévy measure ν. We consider the first passage time Tx of (Xt, t ≥ 0) to level x > 0, and Kx := XTx - x the overshoot and Lx := x- XTx- the undershoot. We first prove that the Laplace transform of the random triple (Tx,Kx,Lx) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that $(\widetilde{T_x},K_x,L_x)$ converges in distribution as x → ∞, where $\widetilde{T_x}$ denotes a suitable renormalization of Tx.