- Tytuł:
- A recurrence theorem for square-integrable martingales
- Autorzy:
- Alsmeyer, Gerold
- Powiązania:
- https://bibliotekanauki.pl/articles/1290203.pdf
- Data publikacji:
- 1994
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Tematy:
-
martingale
stochastic $L_2$-boundedness
recurrence
first passage time
Blackwell's renewal theorem
coupling - Opis:
- 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.
- Źródło:
-
Studia Mathematica; 1994, 110, 3; 221-234
0039-3223 - Pojawia się w:
- Studia Mathematica
- Dostawca treści:
- Biblioteka Nauki
Artykuł