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A recurrence theorem for square-integrable martingales

100%

Alsmeyer G.

Studia Mathematica

1994

tom 110

nr 3

221-234

Let $(M_n)_{n≥0}$ be a zero-mean martingale with canonical filtration $(ℱ_n)_{n≥0}$ and stochastically $L_2$-bounded increments $Y_1,Y_2,..., $ which means that $P(|Y_n| > t | ℱ_{n-1}) ≤ 1 - H(t)$ a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let $V^2 = ∑_{n≥1} E(Y_{n}^{2}|ℱ_{n-1})$. It is the main result of this paper that each such martingale is a.s. convergent on {V < ∞} and recurrent on {V = ∞}, i.e. $P(M_{n} ∈ [-c,c] i.o. | V = ∞) = 1$ 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. As an application of our recurrence theorem, we obtain an extension of Blackwell's renewal theorem to a fairly general class of processes with independent increments and linear positive drift function.

Transient and stationary characteristics of a packet buffer modelled as an MAP/SM/1/b system

100%

Rusek K., Janowski L., Papir Z.

International Journal of Applied Mathematics and Computer Science

2014

tom 24

nr 2

429-442

A packet buffer limited to a fixed number of packets (regardless of their lengths) is considered. The buffer is described as a finite FIFO queuing system fed by a Markovian Arrival Process (MAP) with service times forming a Semi-Markov (SM) process (MAP/SM/1/b in Kendall's notation). Such assumptions allow us to obtain new analytical results for the queuing characteristics of the buffer. In the paper, the following are considered: the time to fill the buffer, the local loss intensity, the loss ratio, and the total number of losses in a given time interval. Predictions of the proposed model are much closer to the trace-driven simulation results compared with the prediction of the MAP/G/1/b model.

Ograniczanie wyników

1 International Journal of Applied Mathematics and Computer Science

1 Studia Mathematica

1 Alsmeyer G.

1 Janowski L.

1 Papir Z.

1 Rusek K.

1 2014

1 1994

Wyniki wyszukiwania

A recurrence theorem for square-integrable martingales

Transient and stationary characteristics of a packet buffer modelled as an MAP/SM/1/b system