TIGHTNESS OF CONTINUOUS STOCHASTIC PROCESSES
Pełen tekst
Having given a probability space P and random variables ξ i : Ω → R m for i = 1, ..., n let us define S k = ξ 1 + ... + ξ k for k = 1, ..., n and S 0 = 0. Then let M n = max 0≤k≤n |S k | and M n0
M n0
M n0
M 20
h=1 [υ h−1 /υ, υ h /υ]. By virtue of the assumption α > 1/2
Let us observe that in particular cases: h = 1 and h = n the above inequalities are trivial, respectively. Similarly as in ([1], Theorem II.12.1) we can verify that M n0
M n0
min +λ1
M n0
M n0
λ 2γ E (u 1 + ... + u n ) 2α
for every λ > 0 and 0 ≤ i < j ≤ n. Then there is a positive number K γ,α0
M n0
λ γ E (u 1 + ... + n n ) α with K γ,α0
≤ K γ,α0
j<δ max−1
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