How Does Tempering Affect the Local and Global Properties of Fractional Brownian Motion?

Divergence of an integral of a process with small ball estimate

“…Then the upper bound (3.11) still holds for B n j in place of B n j , and moreover, on A n j ∩ B n j we have, as before that |f (X t )| ≥ c n on the interval of length at least n −ǫ (2) , otherwise, for sufficiently large n 4) , therefore, for sufficiently large n 1) , and we can conclude as in Theorem 3.1.…”

“…In this case n −ǫ(0) < K 3 n −γǫ (1) , and η n = n −ǫ(0) . Then 1) , s n j = (j − 1)∆ n and t n j = j∆ n for j, n ∈ N. Let…”

“…It means that we are exactly in conditions of Example 2.3 and can produce the same conclusions. [1,15,16]. We shall introduce it as follows.…”

“…Such numbers ǫ(i) (i = 0, 1, 2, 3, 4) exist; for example, let δ ↓ 0 for (1) . The 2nd value, K 3 n −γǫ (1) is necessary in order to apply assumption (iii), the relaxed small ball estimate.…”

“…Such numbers ǫ(i) (i = 0, 1, 2, 3, 4) exist; for example, let δ ↓ 0 for (1) . The 2nd value, K 3 n −γǫ (1) is necessary in order to apply assumption (iii), the relaxed small ball estimate. However, we shall deal with the 1st value, n −ǫ(0) , therefore, assume that n is sufficiently large, such that log n > (1) .…”