Download e-book for kindle: Across Coveted Lands; or, A Journey from Flushing (Holland) by Arnold Henry Savage 1865-1924 Landor

By Arnold Henry Savage 1865-1924 Landor

ISBN-10: 1402194722

ISBN-13: 9781402194726

This Elibron Classics publication is a facsimile reprint of a 1902 version by way of Macmillan & Co., Ltd., London.

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Extra info for Across Coveted Lands; or, A Journey from Flushing (Holland) to Calcutta, Overland: Volume 2

Example text

2) 0 1−2α where gσ (s) = χ[0,σ] (s)(σ − s)−α . Since α < 1/2, gσ ∈ H and |gσ |2 = σ1−2α . 2) is a Riemann integral of the H-valued function (t − σ)α−1 gσ . Using the fact that the mapping H → L2 (H, µ), f → Wf , is continuous, we obtain the following representation formula for B: B(t) = sin πα π t (t − σ)α−1 Wgσ dσ. 2 below. To show summability of Wg(·) (x), we notice that, since Wgσ is a real Gaussian random variable with law N σ1−2α , we have 1−2α |Wgσ (x)|2m µ(dx) = H (2m)! (1 − 2α)−m σ m(1−2α) .

If ϕ ∈ Cb0,1 (H) we set [ϕ]1 := sup x,y∈H x=y |ϕ(x) − ϕ(y)| , |x − y| ϕ ∈ Cb0,1 (H). Cb0,1 (H) is a Banach space with the norm ϕ 1 := ϕ + [ϕ]1 , ϕ ∈ Cb0,1 (H). 0 (v) Cb1,1 (H) is the space of all functions ϕ ∈ Cb1 (H) such that Dϕ is Lipschitz continuous. We set [ϕ]1,1 := sup x=y |Dϕ(x) − Dϕ(y)| , ϕ ∈ Cb1,1 (H). |x − y| Cb1,1 (H) is a Banach space with the norm ϕ 1,1 = f 1 + [ϕ]1,1 , ϕ ∈ Cb1,1 (H). (vi) Cbα (H), α ∈ (0, 1), is the subspace of Cb (H) of all functions ϕ : H → R such that [ϕ]1 := sup x,y∈H x=y |ϕ(x) − ϕ(y)| , |x − y|α ϕ ∈ Cb0,1 (H).

Eλ dµ G dλ Eλ dν G dλ 1/2 dµ dλ ≥ Eλ 6 1/2 dν dλ 1/2 G . 3) Eλ (η|G) is the conditional expectation of the random variable η with respect to G and measure λ. 7 For positive numbers a, b, c, d, ab cd ≤ 1 2 a c + b d . 3), the required result follows. Now let us consider two sequences of measures (µk ) and (νk ) on (R, B(R)) such that νk ∼ µk for all k ∈ N. We set λk = 12 (µk + νk ), and we consider the Hellinger integral H(µk , νk ) = dµk dνk (x) (x) dλk dλk R 1/2 λk (dx), k ∈ N. 4 Since (µk ) and (νk ) are equivalent, we have dµk dνk dµk dνk dµk dνk = = dλk dλk dλk dµk dλk dµk dµk dλk 2 .

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Across Coveted Lands; or, A Journey from Flushing (Holland) to Calcutta, Overland: Volume 2 by Arnold Henry Savage 1865-1924 Landor


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