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Get Invariant Random Fields on Spaces with a Group Action PDF

By Anatoliy Malyarenko, Nicolai Leonenko

ISBN-10: 3642334059

ISBN-13: 9783642334054

ISBN-10: 3642334067

ISBN-13: 9783642334061

The writer describes the present cutting-edge within the conception of invariant random fields. This idea is predicated on a number of assorted parts of arithmetic, together with likelihood conception, differential geometry, harmonic research, and unique services. the current quantity unifies many effects scattered in the course of the mathematical, actual, and engineering literature, in addition to it introduces new effects from this zone first proved via the writer. The booklet additionally provides many useful functions, specifically in such hugely attention-grabbing parts as approximation thought, cosmology and earthquake engineering. it really is meant for researchers and experts operating within the fields of stochastic approaches, information, practical research, astronomy, and engineering.

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36 2 Spectral Expansions It follows that the measure μ is SO(∞)-invariant. By the result of Umemura (1965), μ has the following form: ∞ μ(p) = μλ (p) dν(λ), 0 where μλ is the product of countably many Gaussian measures on R with zero mean and variance λ, while ν is a finite measure on [0, ∞). Therefore, we have R(x, y) = = R∞ ∞ R∞ ∞ dμλ (p) dν(λ) 0 0 = ∞ ei(p,x−y) ei(p,x−y) dμλ (p) dν(λ) exp −λ x − y 2 /2 dν(λ). 0 The rightmost side of the above display is continuous in the Hilbert space topology.

46) 0 The integrand may be written as exp −λ x − y 2 /2 = exp −λ x 2 /2 exp −λ x 2 /2 exp λ(x, y) . By Taylor expansion ∞ exp λ(x, y) = exp(λxj yj ) j =1 ∞ ∞ = j =1 k=0 (λxj yj )k k! ,km )∈Lm λm k1 ! · · · km ! m (xi yi )k , =1 where L0 := ∅, Lm , m ≥ 1 is the set of all multi-indices = (i1 , . . , im , k1 , . . , km ) where i1 , . . , im are positive integers with 1 ≤ i1 < · · · < im , k1 , . . , km are positive integers with k1 + · · · + km = m. 3 Invariant Random Fields on Homogeneous Spaces 37 where xik1 · · · xikmm f (x, λ, m) = exp −λ x 2 /2 λm/2 √ 1 .

Then its mean value is (V ) E Xj (t) = E[Z11 0 ], 0, ˆ K (W ), V0 ∈ G otherwise, ˆ K (W ) iff W is trivial (by Frobenius reciprocity, which is constant. 6). 53) is Rjj (t1 , t2 ) dim V dim V dim W (V ) = (V ) E Zmn Zm n (W YV m )j (t1 )(W YV m )j (t2 ) ˆ K (W ) m=1 m =1 n,n =1 V ,V ∈G dim V dim W (V ) = Rnn ˆ K (W ) n,n =1 V ∈G = 1 dim W (W YV m )j (t1 )(W YV m )j (t2 ) m=1 dim W Rnn Vp+j,p+j g1−1 g2 , (V ) dim V n,n =1 ˆ K (W ) V ∈G where g1 (resp. g2 ) is an arbitrary element from the left coset corresponding to t1 (resp.

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Invariant Random Fields on Spaces with a Group Action by Anatoliy Malyarenko, Nicolai Leonenko


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