# BRWS

AcronymDefinition
BRWSBrowser Protocol Specification
BRWSBengal Rural Welfare Service (India)
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and call [([[PI]].sub.n]).sub.n [greater than or equal to] 0] a BRW on R with offspring distribution [([p.sub.j]).sub.j [greater than or equal to] 0] and increment distribution Q.
Definition 2.1 A BRW [([[PI].sub.n]).sub.n [greater than or equal to] 0] with increment distribution Q is called d-arithmetic if Q is d-arithmetic, i.e., if
Such a Q as well as an associated BRW is called genuinely two-sided hereafter.
Proposition 2.2 Let [([[PI].sub.n]).sub.n [greater than or equal to] 0] be a genuinely two-sided, d-arithmetic BRW, d [member of] {0, 1}, with increment distribution Q.
Definition 2.4 (a) A genuinely two-sided 1-arithmetic BRW [([[PI].sub.n]).sub.n [greater than or equal to] 0] is called recurrent if
(b) A genuinely two-sided nonarithmetic BRW [([[PI].sub.n]).sub.n [greater than or equal to] 0] is called (topologically) recurrent if
Theorem 2.5 Let [([[PI].sub.n]).sub.n [greater than or equal to] 0] be a genuinely two-sided, d-arithmetic BRW, d [member of] {0, 1}, with increment distribution Q.
In view of this result, the BRW [([[PI].sub.n]).sub.n [greater than or equal to] 0] is called critical, if m[PSI](v) = 1, subcritical, if m[PSI](v) < 1, and supercritical, if m[PSI](v) > 1.
Proposition 2.2 then ensures that this probability must be 0 as claimed, in other words, the BRW is recurrent.
The following argument embarks on Lemma 4.3 by which it suffices to consider the function H so as to assess recurrence or transience of the given BRW By Lemma 4.2 and in the notation from there, this can be done by computing the mean offspring [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the Galton-Watson process [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Once knowing that the critical BRW is transient and thus drifting to [infinity], it is natural ask for its minimal speed or, equivalently, the asymptotic behavior of the leftmost particle in the cloud as time goes to infinity.
Theorem 5.1 Let [([[PI].sub.n]).sub.n[greater than or equal to]0] be a genuinely two-sided critical BRW satisfying [p.sub.0] = 0, [p.sub.1] < 1, [mu](Q) [member of] (0, [infinity]) and (11).
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