UNIFORM UPPER BOUND FOR A STABLE MEASURE OF A SMALL BALL

The authors of [1] stated the following conjecture: Let (cid:22) be a symmetric (cid:11) -stable measure on a separable Banach space and B a centered ball such that (cid:22) ( B ) (cid:20) b . Then there exists a constant R ( b ) , depending only on b , such that (cid:22) ( tB ) (cid:20) R ( b ) t(cid:22) ( B ) for all 0 < t < 1 . We prove that the above inequality holds but the constant R must depend also on (cid:11) . Recently, Let (cid:22) be a symmetric -stable measure, < 2, a separable Banach space, b < 1, and let B denote a centered ball such that B ) b . Then there exists a depending , t , quantity t (cid:11)= 2 t Conjecture

Recently, the authors of [1] proved the following (Theorem 6.4 in [1]): Let µ be a symmetric α-stable measure, 0 < α ≤ 2, on a separable Banach space, fix b < 1, and let B denote a centered ball such that µ(B) ≤ b. Then there exists a constant R(b) = 3 b √ 1−b , depending only on b, such that for all 0 ≤ t ≤ 1 µ(tB) ≤ R(b)t α/2 µ(B). (1) Of course, for small values of t, the quantity t α/2 is much larger than t. The authors of [1] stated in their Conjecture 7.4 that (1) is true for all symmetric α-stable measures with t instead of t α/2 and some R(b) depending only on b.
In our earlier paper [3], we also gave an estimate of a stable measure of a small ball. Namely, we proved the following. Let µ be a symmetric α-stable measure, 0 < α ≤ 2, on a separable Banach space, put B = {x : x ≤ 1}, let 0 < r < α and suppose that µ is so normalized that x r µ (dx) = 1. Then there exists a constant K = K(α, r) such that for all 0 ≤ t ≤ 1 µ(tB) ≤ K(α, r) t.
(2) Some estimates of K(α, r) were also given in [3], we recall one of them in the final Remark. Some normalization of µ is needed, as we will show in the sequel (see Example), in the paper [3] we chose the normalizing condition x r µ (dx) = 1. But proving the inequality (2), we also obtained the inequality In this note we will show that using (3) we can prove an estimate that is very close to the above-mentioned conjecture, however, the constant R(b) must depend also on α.
The following is a generalization of (1).
First we show that the constant R must depend on α.
Example. Suppose that there exists positive function R(b) that fulfills (4), does not depend on α and is bounded on every closed subinterval of (0, 1) Let X α be an α-stable random variable with the characteristic function e −|t| α . It is known (see e.g. [4]) that where W is a random variable having the exponential distribution with mean 1. Consider Denote by µ the distribution of X α . It is easy to compute the value of the density of µ at zero: contradicting the inequality (4).

Stable Measure of a Small Ball 77
This implies that R(b) must also depend on α.
The proof of the theorem is almost the same as the proof of (1) in the paper [1], the difference is that instead of Kanter inequality we use our estimate (3). For the sake of completeness we repeat this proof. We start with two lemmas.
Proof of the Theorem. Fix B with µ(B) ≤ b and take s ≥ 1 such that µ(sB) = b. Now, in Lemma 2, put κ = t and t = 1 2s . Then Taking different values of r ∈ (0, α) we get different values of K(α, r). If, for simplicity, we take r = α/2 we get R(α, b) = K(α, α/2) Remark. Let us recall some estimates of K(α, r) which were given in the paper [3]. If we take r = α/2 then where Φ is the distribution function of a standard normal variable. For different values of r other estimates are possible, it could be interesting to find the least value of K(α, r). Of course, if we consider α ≥ ε > 0 then we can find