1. Introduction.

Consider a Markov process Y taking values in a Polish space,

M M

E. For M€ IN, K(M) particles start at xx (u) , . . . ,xR,.- . (J) € E and

follow independent copies of Y on the time interval [0,1/M). At t

= 1/M each particle dies or splits into two, each with probability

1/2, independently of each other- The new individuals then follow

independent copies of Y on [L/M,2/M), and this pattern of

alternating spatial motions and branching mechanisms continues

indefinitely. In this way we construct a random "Y-tree" of

branching particles. Define a random measure on (E,&) (& is the

Borel cr-field) by

X^(u)(A) = M"1(no. of particles in A at time t), Ae&, t^O.

is a process taking values in the space M„(E) of finite

measures on E with the topology of weak convergence. Watanabe

(1968) (see Ethier-Kurtz (1986, Ch. 9) for the required tightness)

showed that if Y is a Feller process and E is locally compact,

then X^ —- mQ (in Mp(E)) implies XM ^-X on D( [0,«) ,Mp(E) ) (weak

convergence of laws), where X is a M^fE)-valued diffusion starting

r

at mn. Let Q denote the limiting law on the appropriate space

Research of both authors partially supported by the Natural

Sciences and Engineering Research Council of Canada.

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