Ecole d'Ete de Probabilites de Saint-Flour VI-1976 by Hoffman-Jorgensen J., Ligget T.M., Neveu J.

By Hoffman-Jorgensen J., Ligget T.M., Neveu J.

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Additional resources for Ecole d'Ete de Probabilites de Saint-Flour VI-1976

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The idea behind the proof is simple: Whenever X has a “big” jump guaranteed by An , this jump corresponds to the jump of Ln+ and then it is very difficult for a spectrally positive process Ln+ to come down, which is required by Bn+,c . 3: on {Ln+ ≥ 21−η¯ c n n1/(1+β )−ε } we may still have Zt (c2−n−2 , 0) ≤ 2−η¯ c n n1/(1+β )−ε if Bn−,c occurs. In other words, this is the situation where the jump of X, which leads to a large value of Ln+ , can be compensated by a further jump of X. The next lemma states that it cannot happen that all the jumps guaranteed by An ’s will be compensated.

24) then pt−s c (t − s)1/2 − y ≤ pt−s c ( j + 1/2)(t − s)1/2 = (t − s)−1/2 p1 c ( j + 1/2) = (2π )−1/2 (t − s)−1/2 e−c 2 (1/2+ j)2 /2 . From this bound we conclude that | y−c (t−s)1/2 |> 2c (t−s)1/2 dy pt−s c (t − s)1/2 − y 1B2 (0) (y)Xs (y) ∞ ≤ (2π )−1/2 (t − s)−1/2 ∑ e−c j=0 2 (1/2+ j)2 /2 Ds, j dy1B2 (0) (y)Xs (y). 24) is the union of two balls of radius c(t − s)1/2 . If furthermore Ds, j ∩ B2 (0) = 0, / then, in view of the assumption c(t − s)1/2 ≤ 1, the centers of those balls lie in B3 (0).

Let ε > 0 and γ ∈ 0, (1+ β )−1 . 14) (t − s)|x| for some s < t and x ∈ B2 (0) ( fs,x ) for some s < t, x ∈ B2 (0) . 20) (t − s)|x| 1 1+β −γ for some s < t and x ∈ B2 (0) ≤ ε . Introduce the event Dθ := Xt (0) > θ , sup Xs (R) ≤ θ −1 , WB3 (0) ≤ θ −1 . 21) 0 0. Proof. 1, to show that the number of jumps is greater than zero almost surely on some event, it is enough to show the divergence of a certain integral on that event or even on a bigger one.

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