On the exit time from a cone for random walks with drift

Random Walks with Drift - a Sequential Approach

In this paper sequential monitoring schemes to detect nonparametric drifts are studied for the random walk case. The procedure is based on a kernel smoother. As a by-product we obtain the asymptotics of the Nadaraya-Watson estimator and its associated sequential partial sum process under non-standard sampling. The asymptotic behavior differs substantially from the stationary situation, if there...

On the Exit Time of a Random Walk with Positive Drift

We study a random walk with positive drift in the first quadrant of the plane. For a given connected region C of the first quadrant, we analyze the number of paths contained in C and the first exit time from C. In our case, region C is bounded by two crossing lines. It is noted that such a walk is equivalent to a path in a tree from the root to a leaf not exceeding a given height. If this tree ...

Exit times for Integrated Random Walks

We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time n. Assuming that the moment of order 2 + δ is finite, we show that the exact asymptotics for this probability are n−1/4. To show these asymptotics we develop a discrete potential theory for integrated random walks.

Random Walks on Trees Withfinitely Many Cone

Conditioned Limit Theorems for Random Walks with Negative Drift*

In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if Sn is a random walk with negative mean and finite variance then there is a constant c~ so that (St,.j/c~nl/2[N>n) converges weakly to a process which he called the Brownian excursion. It will be shown that his conjecture is false or, more precisely, that if ES~ = a < 0 , ES~ < oo, and there is a slowly var...