Probability: Theory and Examples.

Durrett, Rick

doi:10.2307/2532227

Cited by 2,077 publications

(3,358 citation statements)

References 0 publications

Supporting

Mentioning

3,299

Contrasting

Unclassified

Order By: Relevance

“…By the Lindeberg-Feller central limit theorem for martingale differences (see, for example, p. 414 from [13]), it is enough to verify that…”

Section: Appendixmentioning

confidence: 99%

Plot of CLT

Jiang

2019

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let Γn be an n × n Haar-invariant orthogonal matrix. Let Zn be the p × q upper-left submatrix of Γn, where p = pn and q = qn are two positive integers. Let Gn be a p × q matrix whose pq entries are independent standard normals. In this paper we consider the distance between √ nZn and Gn in terms of the total variation distance, the Kullback-Leibler distance, the Hellinger distance and the Euclidean distance. We prove that each of the first three distances goes to zero as long as pq/n goes to zero, and not so if (p, q) sits on the curve pq = σn, where σ is a constant. However, it is different for the Euclidean distance, which goes to zero provided pq 2 /n goes to zero, and not so if (p, q) sits on the curve pq 2 = σn. A previous work by Jiang [17] shows that the total variation distance goes to zero if both p/ √ n and q/ √ n go to zero, and it is not true provided p = c √ n and q = d √ n with c and d being constants. One of the above results confirms a conjecture that the total variation distance goes to zero as long as pq/n → 0 and the distance does not go to zero if pq = σn for some constant σ.

show abstract

“…By the Lindeberg-Feller central limit theorem for martingale differences (see, for example, p. 414 from [13]), it is enough to verify that…”

Section: Appendixmentioning

confidence: 99%

Plot of CLT

Jiang

2019

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…To complete the proof of Theorem 2.1, we apply Slutsky's Theorem [1] which states that if X n , Y n , W n are sequences of random variables, and X n P − → a, Y n P − → b, (where a, b are constants) and W n D − → W (for some random variable W ) then…”

mentioning

confidence: 99%

Distribution of phylogenetic diversity under random extinction

Faller¹,

Pardi²,

Steel³

2008

Journal of Theoretical Biology

View full text Add to dashboard Cite

Abstract. Phylogenetic diversity is a measure for describing how much of an evolutionary tree is spanned by a subset of species. If one applies this to the (unknown) subset of current species that will still be present at some future time, then this 'future phylogenetic diversity' provides a measure of the impact of various extinction scenarios in biodiversity conservation. In this paper we study the distribution of future phylogenetic diversity under a simple model of extinction (a generalized 'field of bullets' model). We show that the distribution of future phylogenetic diversity converges to a normal distribution as the number of species grows (under mild conditions, which are necessary). We also describe an algorithm to compute the distribution efficiently, provided the edge lengths are integral, and briefly outline the significance of our findings for biodiversity conservation.

show abstract

“…Further it can be shown that this critical probability p c = 1/2 on a 2-D grid (d = 2). This seemingly simple result almost took a decade to be established [19] and transition thresholds for percolation models in general are unknown [13]. As such researchers usually provide empirical evidence for results using MonteCarlo simulations where strong theoretical results are not yet available.…”

Section: B Preliminariesmentioning

confidence: 99%

“…Firstly, although growth models on lattices forms the basis of percolation theory literature [11]- [13], designing and analyzing feedback polices to control growth process is novel. Secondly, it is important to note that in the related literature on information, rumor, and epidemic spreading on networks [14]- [17], optimal policies to control epidemics on general networks are not known [18].…”

mentioning

confidence: 99%

Controlling stochastic growth processes on lattices: Wildfire management with robotic fire extinguishers

Somanath

Karaman

2014

53rd IEEE Conference on Decision and Control

View full text Add to dashboard Cite

Abstract-Forest fires continue to cause considerable social and economic damage. Fortunately, the emergence of new robotics technologies, including capable autonomous unmanned aerial vehicles, may help improve wildfire management in the near future. In this paper, we characterize the number of vehicles required to combat wildfires, using a percolation-theoretic analysis that originated in the mathematical physics community. We model the wildfire as a stochastic growth process on a square lattice, where the local growth probabilities depend on the presence of robotic fire-extinguishing vehicles. We develop two control policies: First treats only a fraction of burning nodes at a given time, and the second treats burning nodes only at finite time intervals. We characterize the conditions under which these policies can stabilize a wildfire, i.e., ensure the fire stops eventually almost surely. We also provide computational results which demonstrate our theoretical analysis.

show abstract

Probability: Theory and Examples.

Cited by 2,077 publications

References 0 publications

Plot of CLT

Plot of CLT

Distribution of phylogenetic diversity under random extinction

Controlling stochastic growth processes on lattices: Wildfire management with robotic fire extinguishers

Contact Info

Product

Resources

About