## Limit theorems for certain branching random walks on compact groups and homogeneous spaces.(English)Zbl 0544.60022

Let G denote an Abelian compact group with Haar measure m. Consider random variables $$X_ 0$$ and $$Y_ 1,Y_ 2,..$$. taking values in G and random variables $$I_ 1,I_ 2,..$$. such that each $$I_ n$$ takes its values in the set $$A_ n:=\{0,1,...,n-1\}$$ ($$n\geq 1)$$. Assume that these random variables are all independent, that $$X_ 0$$ has distribution $$\nu$$ on G, each $$Y_ n$$ has distribution $$\mu$$ on G and each $$I_ n$$ a uniform distribution on $$A_ n$$ ($$n\geq 1)$$. Define inductively the sequence $$X_ 1,X_ 2,..$$. of random variables $$X_ n:=X_{I_ n}\cdot Y_ n$$ ($$n\geq 1)$$. The branching random walk $$(X_ n)_{n\geq 1}$$ on G reduces to the underlying simple random walk on G if one puts $$I_ n=n-1$$. It is further assumed that $$\mu$$ is not supported by any proper closed subgroup of G.
At first the author notes that the sequence $$(P_{X_ n})_{n\geq 1}$$ of distributions $$P_{X_ n}$$ of $$X_ n$$ converges weakly to m. (Extension of the Kawada-Ito theorem to branching random walks). Thus the products $$X_ 0\cdot...\cdot X_ n$$ will be spread out over G for large n. The author’s aim in the paper under review is to study the fluctuations from m, i.e. the asymptotic behavior in distribution of the sequences $$(S_ n(f))_{n\geq 1}$$ of sums $$S_ n(f):=\sum^{m}_{j=0}f(X_ j)$$ for $$f\in L^ 2(G,m)$$. In the special case $$\mu:=m$$ the sequence $$(S_ n(f))_{n\geq 1}$$ is asymptotically normally distributed.
For arbitrary $$\mu$$ and real characters $$\gamma$$ of G it is shown that, if the Fourier transform $${\hat \mu}$$($$\gamma)$$ of $$\gamma$$ has real part $$<{1\over2}$$, the sequence $$(n^{-1/2}S_ n(\gamma))_{n\geq 1}$$ converges in distribution to the normal random variable N(0,(1-2$${\hat \mu}$$($$\gamma))$$$${}^{-1})$$ as $$n\to \infty.$$
Although the proof of this result is already rather technical, the author also achieves asymptotic results for arbitrary complex characters, under the assumption Re $${\hat \mu}$$($$\gamma)$$$$={1\over2}$$ (with the norming (n log n)$${}^{-1/2}$$ instead of $$n^{-1/2})$$, or more general functions $$f\in L^ 2(G,m)$$ instead of characters $$\gamma$$ of G, for arbitrary compact groups G, and finally for compact homogeneous spaces G/K with respect to compact subgroups K of G. In the latter case, however, the definition of a branching random walk has to be modified appropriately. A list of very interesting examples ranging from the Pólya-Eggenberg urn model to genetic branching on the sphere closes the paper.
Reviewer: H.Heyer

### MSC:

 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60G50 Sums of independent random variables; random walks 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60F05 Central limit and other weak theorems
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