A counterpart of the Borel-Cantelli lemma

1980 ◽  
Vol 17 (04) ◽  
pp. 1094-1101 ◽  
Author(s):  
F. Thomas Bruss
Keyword(s):  
Random Walk ◽  
Probability Space ◽  
Branching Processes ◽  
Simple Random Walk ◽  
Cantelli Lemma ◽  
Sequence Of Events ◽  
Independence Condition ◽  
General Part

The general part of the Borel-Cantelli lemma says that for any sequence of events (An) defined on a probability space (Ω, Σ,P), the divergence of ΣnP(An) is necessary forP(Ani.o.) to be one (see e.g. [1]). The sufficient direction is confined to the case where the Anare independent. This paper provides a simple counterpart of this lemma in the sense that the independence condition is replaced byfor some. We will see that this property of (An) may frequently be assumed without loss of generality. We also disclose a useful duality which allows straightforward conclusions without selecting independent sequences. A simple random walk example and a new result in the theory ofϕ-branching processes will show the tractability of the method.

A counterpart of the Borel-Cantelli lemma

1980 ◽  
Vol 17 (4) ◽  
pp. 1094-1101 ◽  
Author(s):  
F. Thomas Bruss
Keyword(s):  
Random Walk ◽  
Probability Space ◽  
Branching Processes ◽  
Simple Random Walk ◽  
Cantelli Lemma ◽  
Sequence Of Events ◽  
Independence Condition ◽  
General Part

The general part of the Borel-Cantelli lemma says that for any sequence of events (An) defined on a probability space (Ω, Σ, P), the divergence of ΣnP(An) is necessary for P(An i.o.) to be one (see e.g. [1]). The sufficient direction is confined to the case where the An are independent. This paper provides a simple counterpart of this lemma in the sense that the independence condition is replaced by for some . We will see that this property of (An) may frequently be assumed without loss of generality. We also disclose a useful duality which allows straightforward conclusions without selecting independent sequences. A simple random walk example and a new result in the theory of ϕ -branching processes will show the tractability of the method.

scholarly journals Diameter and Stationary Distribution of Random $r$-Out Digraphs

10.37236/9485 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Louigi Addario-Berry ◽  
Borja Balle ◽  
Guillem Perarnau
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Stationary Distribution ◽  
High Probability ◽  
Branching Processes ◽  
Simple Random Walk ◽  
Undirected Graphs

Let $D(n,r)$ be a random $r$-out regular directed multigraph on the set of vertices $\{1,\ldots,n\}$. In this work, we establish that for every $r \ge 2$, there exists $\eta_r>0$ such that $\mathrm{diam}(D(n,r))=(1+\eta_r+o(1))\log_r{n}$. The constant $\eta_r$ is related to branching processes and also appears in other models of random undirected graphs. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on $D(n,r)$. In particular, we determine the asymptotic behaviour of $\pi_{\max}$ and $\pi_{\min}$, the maximum and the minimum values of the stationary distribution. We show that with high probability $\pi_{\max} = n^{-1+o(1)}$ and $\pi_{\min}=n^{-(1+\eta_r)+o(1)}$. Our proof shows that the vertices with $\pi(v)$ near to $\pi_{\min}$ lie at the top of "narrow, slippery tower"; such vertices are also responsible for increasing the diameter from $(1+o(1))\log_r n$ to $(1+\eta_r+o(1))\log_r{n}$.

The size of the region occupied by one type in an invasion process

1976 ◽  
Vol 13 (02) ◽  
pp. 355-356 ◽  
Author(s):  
Aidan Sudbury
Keyword(s):  
Random Walk ◽  
Simple Random Walk ◽  
Rectangular Lattice ◽  
Invasion Process

Particles are situated on a rectangular lattice and proceed to invade each other's territory. When they are equally competitive this creates larger and larger blocks of one type as time goes by. It is shown that the expected size of such blocks is equal to the expected range of a simple random walk.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

A Simple Random Walk Model for Dairy Cow Calving Time Prediction

2021 ◽  
Author(s):  
Thi Thi Zin ◽  
Pyke Tin ◽  
Pann Thinzar Seint ◽  
Kosuke Sumi ◽  
Ikuo Kobayashi ◽  
...  
Keyword(s):  
Random Walk ◽  
Dairy Cow ◽  
Simple Random Walk ◽  
Random Walk Model ◽  
Time Prediction

scholarly journals Diamond aggregation

2010 ◽  
Vol 149 (2) ◽  
pp. 351-372
Author(s):  
WOUTER KAGER ◽  
LIONEL LEVINE
Keyword(s):  
Random Walk ◽  
Growth Model ◽  
Power Law ◽  
Simple Random Walk ◽  
Internal Diffusion ◽  
Diffusion Limited Aggregation ◽  
Directional Bias ◽  
Model Based ◽  
Diffusion Limited ◽  
Logarithmic Fluctuations

AbstractInternal diffusion-limited aggregation is a growth model based on random walk in ℤd. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in ℤ2 for which the limiting shape is a diamond. Certain of these walks—those with a directional bias toward the origin—have at most logarithmic fluctuations around the limiting shape. This contrasts with the simple random walk, where the limiting shape is a disk and the best known bound on the fluctuations, due to Lawler, is a power law. Our walks enjoy a uniform layering property which simplifies many of the proofs.

On the number of times where a simple random walk reaches its maximum

1992 ◽  
Vol 29 (02) ◽  
pp. 305-312 ◽  
Author(s):  
W. Katzenbeisser ◽  
W. Panny
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Probability And Statistics

Let Qn denote the number of times where a simple random walk reaches its maximum, where the random walk starts at the origin and returns to the origin after 2n steps. Such random walks play an important role in probability and statistics. In this paper the distribution and the moments of Qn , are considered and their asymptotic behavior is studied.

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