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Abstract:
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Let M be a maximum and let N be a minimum of the non-negative martingale
X1, X2, . . . , Xn. It is well known, that if X1 = 1, then
γ(‖M ‖1) ≤ E (Xn log Xn) and γ(‖N ‖1) ≤ E (Xn log Xn) ,
where γ(x) = x − 1 − log x, for all x > 0. In this thesis, we prove the analogue of this result in the
case when 1 < p < ∞, by proving that
δp
(‖M ‖p
p
) ≤ ‖Xn‖p and δp
(‖N ‖p
p
) ≤ ‖Xn‖p,
where δp(x) =
(
1 − 1
p
)
x 1
p + 1
p x 1
p −1, for all x > 0. We also obtain a probabilistic proof of the fact
min
ρ∈D(Qn)
∫
Qn
dx1 . . . dxn
ρ (x1, . . . , xn)p−1 ∏n
j=1 xαj +1
j
=
n∏
j=1
( p
p − αj − 1
)p
,
where p > 1, αj < p − 1 for j = 1, . . . , n and D (Qn) is family of all densities on the n-dimensional unit
cube Qn = (0, 1)n in Rn. This provides the proof of the multidimensional weighted Hardy inequality.
Namely, if f : Rn
+ → (0, ∞) is a measurable function, p > 1 and αj < p − 1 for j = 1, . . . , n, then
∫
Rn
+
n∏
j=1
xαj
j Hnf (x)p dx ≤
n∏
j=1
( p
p − αj − 1
)p ∫
Rn
+
n∏
j=1
xαj
j f (x)p dx,
where
Hnf (x) = 1
x1 . . . xn
∫ x1
0
· · ·
∫ xn
0
f (t) dt,
is a multidimensional Hardy operator, x = (x1, . . . , xn) ∈ Rn
+, t = (t1, . . . , tn) and dt = dt1 . . . dtn.
Let B(t) be a standard planar Brownian motion and r(θ) be the length of the projection of B[0, 1]
on the line generated by the unit vector eθ = (cos θ, sin θ), where 0 ≤ θ ≤ π. We nd the common
distribution function F of the random variables r(θ). Namely, we prove that
F(x) = 8
∞∑
n=1
( 1
x2 + 1
(2n − 1)2π2
)
exp
(
− (2n − 1)2π2
2x2
)
,
for every x > 0. As immediate consequence, lower bound for the expected diameter of the set B[0, 1],
better than known, is obtained. Namely, it is known that Ed ≥ 1.601, where d is the diameter of the
set B[0, 1]. In this thesis we show Ed ≥ 1.856. |