J. Biggins

Publications69
h-index26
Citations2,664
Highly Influential Citations341
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MARTINGALE CONVERGENCE IN THE BRANCHING RANDOM WALK
A result like the Kesten-Stigum theorem is obtained for certain martingales associated with the branching random walk. A special case, when a 'Malthusian parameter' exists, is considered in greater…
Measure change in multitype branching
The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its…
Uniform Convergence of Martingales in the Branching Random Walk
THE FIRST- AND LAST-BIRTH PROBLEMS FOR A MULTITYPE AGE-DEPENDENT BRANCHING PROCESS
If B" is the time of the first birth in the nth generation in a supercritical irreducible multitype Crump-Mode process then when there are people in every generation B"/n converges to a constant; if…
Fixed Points of the Smoothing Transform: the Boundary Case
Let $A=(A_1,A_2,A_3,\ldots)$ be a random sequence of non-negative numbers that are ultimately zero with $E[\sum A_i]=1$ and $E \left[\sum A_{i} \log A_i \right] \leq 0$. The uniqueness of the…
SENETA-HEYDE NORMING IN THE BRANCHING RANDOM WALK
!. !. !. malization of the general Crump! Mode! Jagers branching process is obtained; in this case the convergence holds almost surely. The results rely heavily on a detailed study of the functional…
Chernoff's theorem in the branching random walk
The Growth and Spread of the General Branching Random Walk
A general (Crump-Mode-Jagers) spatial branching process is considered. The asymptotic behavior of the numbers present at time t in sets of the form [ ta, oco) is obtained. As a consequence it is…
LARGE DEVIATIONS IN THE SUPERCRITICAL BRANCHING PROCESS
The tail behaviour of the limit of the normalized population size in the simple supercritical branching process, W, is studied. Most of the results concern those cases when a tail of the distribution…
HOW FAST DOES A GENERAL BRANCHING RANDOM WALK SPREAD
New results on the speed of spread of the one-dimensional spatial branching process are described. Generalizations to the multitype case and to d dimensions are discussed. The relationship of the…
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