Two-Parameter Stochastic Processes With Finite Variation
Two-Parameter Stochastic Processes With Finite Variation
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In our eReader you can find the full English version of the book. Read Two-Parameter Stochastic Processes With Finite Variation Online - link to read the book on full screen. Our eReader also allows you to upload and read Pdf, Txt, ePub and fb2 books. In the Mini eReder on the page below you can quickly view all pages of the book - Read Book Two-Parameter Stochastic Processes With Finite Variation
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Let R , Re 6, disjoint with R < U R ? e *• Now » R i ^ R p is a ^ s0 a rectangle iff they "match up" on one side (see Figure 3~1). I I I t" R 2 t' R l 1 t s c ' Figure 3~1 Additivity on 5 71 Denote R = (s,s'] x (t,t'] R 2 = (s,s'] x (t',t"l (Figure 3 _ 1). (The proof is the same if R is of the forr (s',s"] x (t,t'].) We have m(R 1 UR 2 ) = f(s"+,t"+) - f(s+,t"+) - f(s'+,t+) + f(s+,t + ) = [f(s'+,t"+) - f(s+,t M +) - f(s'+,t"+) + f(s+,t'+)] + [f(s'+,t'+) - f(s+,t'+) - f(s'+,t+) + f(s+,t+)] (we added and subtracted f(s'+,t'+) and f(s+,t'+)) = m(R ) + m(R ). 3) o has finite variation on 6. We prove this by contradiction; suppose there exists a rectangle J e 6 such that |o|(J) = ♦ ■ (|a| denotes the variation of o). Then, for each a>0, there exists a finite family (J ), h = 1,2, ...n of disjoint rectangles from 6, J C j for all h, such that h n n I |o(J ) | > a, i.e., I |A (f+) | > a. h-1 n h-1 n Denote J = ( (s. ,t. ) ,(s ',t ')] for all h. We may, of course, assume h h h h h n that J = \^J J (so that [^J j c 5).
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To quickly assess the difficulty of the text, read a short excerpt:
Let R , Re 6, disjoint with R < U R ? e *• Now » R i ^ R p is a ^ s0 a rectangle iff they "match up" on one side (see Figure 3~1). I I I t" R 2 t' R l 1 t s c ' Figure 3~1 Additivity on 5 71 Denote R = (s,s'] x (t,t'] R 2 = (s,s'] x (t',t"l (Figure 3 _ 1). (The proof is the same if R is of the forr (s',s"] x (t,t'].) We have m(R 1 UR 2 ) = f(s"+,t"+) - f(s+,t"+) - f(s'+,t+) + f(s+,t + ) = [f(s'+,t"+) - f(s+,t M +) - f(s'+,t"+) + f(s+,t'+)] + [f(s'+,t'+) - f(s+,t'+) - f(s'+,t+) + f(s+,t+)] (we added and subtracted f(s'+,t'+) and f(s+,t'+)) = m(R ) + m(R ). 3) o has finite variation on 6. We prove this by contradiction; suppose there exists a rectangle J e 6 such that |o|(J) = ♦ ■ (|a| denotes the variation of o). Then, for each a>0, there exists a finite family (J ), h = 1,2, ...n of disjoint rectangles from 6, J C j for all h, such that h n n I |o(J ) | > a, i.e., I |A (f+) | > a. h-1 n h-1 n Denote J = ( (s. ,t. ) ,(s ',t ')] for all h. We may, of course, assume h h h h h n that J = \^J J (so that [^J j c 5).
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You can find similar books in the "Read Also" column, or choose other free books by Charles Lindsey to read online
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