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MARTINGALE CONVERGENCE IN THE BRANCHING RANDOM WALK

- J. Biggins
- Mathematics
- 1 March 1977

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… Expand

Measure change in multitype branching

- J. Biggins, A. Kyprianou
- MathematicsAdvances in Applied Probability
- 1 June 2004

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… Expand

THE FIRST- AND LAST-BIRTH PROBLEMS FOR A MULTITYPE AGE-DEPENDENT BRANCHING PROCESS

- J. Biggins
- Mathematics
- 1 September 1976

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… Expand

Fixed Points of the Smoothing Transform: the Boundary Case

- J. Biggins, A. Kyprianou
- Mathematics
- 13 June 2005

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… Expand

SENETA-HEYDE NORMING IN THE BRANCHING RANDOM WALK

- J. Biggins, A. Kyprianou
- Mathematics
- 1997

!. !. !. 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… Expand

The Growth and Spread of the General Branching Random Walk

- J. Biggins
- Mathematics
- 1 November 1995

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… Expand

LARGE DEVIATIONS IN THE SUPERCRITICAL BRANCHING PROCESS

- J. Biggins, N. Bingham
- Mathematics
- 1 December 1993

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… Expand

HOW FAST DOES A GENERAL BRANCHING RANDOM WALK SPREAD

- J. Biggins
- Mathematics
- 1997

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… Expand

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