On the local time of a recurrent random walk on ℤ²

We prove a functional limit theorem for the number of visits by a planar random walk on Z 2 \mathbb {Z}^2 with zero mean and finite second moment to the points of a fixed finite set P ⊂ Z 2 P\subset \mathbb {Z}^2 . The proof is based on the analysis of an accompanying random process with immigration at renewal epochs in case when the inter-arrival distribution has a slowly varying tail.

Empirical processes for recurrent and transient random walks in random scenery

In this paper, we are interested in the asymptotic behaviour of the sequence of processes (Wn(s,t))s,t∈[0,1] with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(\mathds{1}_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where (ξx, x ∈ ℤd) is a sequence of independent random variables uniformly distributed on [0, 1] and (Sn)n ∈ ℕ is a random walk evolving in ℤd, independent of the ξ’s. In M. Wendler [Stoch. Process. Appl. 126 (2016) 2787–2799], the case where (Sn)n ∈ ℕ is a recurrent random walk in ℤ such that (n−1/αSn)n≥1 converges in distribution to a stable distribution of index α, with α ∈ (1, 2], has been investigated. Here, we consider the cases where (Sn)n ∈ ℕ is either: (a) a transient random walk in ℤd, (b) a recurrent random walk in ℤd such that (n−1/dSn)n≥1 converges in distribution to a stable distribution of index d ∈{1, 2}.

Neural Collective Entity Linking Based on Recurrent Random Walk Network Learning

Benefiting from the excellent ability of neural networks on learning semantic representations, existing studies for entity linking (EL) have resorted to neural networks to exploit both the local mention-to-entity compatibility and the global interdependence between different EL decisions for target entity disambiguation. However, most neural collective EL methods depend entirely upon neural networks to automatically model the semantic dependencies between different EL decisions, which lack of the guidance from external knowledge. In this paper, we propose a novel end-to-end neural network with recurrent random-walk layers for collective EL, which introduces external knowledge to model the semantic interdependence between different EL decisions. Specifically, we first establish a model based on local context features, and then stack random-walk layers to reinforce the evidence for related EL decisions into high-probability decisions, where the semantic interdependence between candidate entities is mainly induced from an external knowledge base. Finally, a semantic regularizer that preserves the collective EL decisions consistency is incorporated into the conventional objective function, so that the external knowledge base can be fully exploited in collective EL decisions. Experimental results and in-depth analysis on various datasets show that our model achieves better performance than other state-of-the-art models. Our code and data are released at https://github.com/DeepLearnXMU/RRWEL.

On the non-recurrent random walk in a random environment

For weakly transient random walk in a random environment that tend at −∞ the limit theorem for the time of hitting a high level is proved.

Microblog Sentiment Classification via Recurrent Random Walk Network Learning

Microblog Sentiment Classification (MSC) is a challenging task in microblog mining, arising in many applications such as stock price prediction and crisis management. Currently, most of the existing approaches learn the user sentiment model from their posted tweets in microblogs, which suffer from the insufficiency of discriminative tweet representation. In this paper, we consider the problem of microblog sentiment classification from the viewpoint of heterogeneous MSC network embedding. We propose a novel recurrent random walk network learning framework for the problem by exploiting both users’ posted tweets and their social relations in microblogs. We then introduce the deep recurrent neural networks with random-walk layer for heterogeneous MSC network embedding, which can be trained end-to-end from the scratch. Weemploytheback-propagationmethodfortraining the proposed recurrent random walk network model. The extensive experiments on the large-scale public datasets from Twitter show that our method achieves better performance than other state-of-the-art solutions to the problem.

Recurrence and Transience of Critical Branching Processes in Random Environment with Immigration and an Application to Excited Random Walks

We establish recurrence and transience criteria for critical branching processes in random environments with immigration. These results are then applied to the recurrence and transience of a recurrent random walk in a random environment on ℤ disturbed by cookies inducing a drift to the right of strength 1.

Recurrence and Transience of Critical Branching Processes in Random Environment with Immigration and an Application to Excited Random Walks

We establish recurrence and transience criteria for critical branching processes in random environments with immigration. These results are then applied to the recurrence and transience of a recurrent random walk in a random environment on ℤ disturbed by cookies inducing a drift to the right of strength 1.