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**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.

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