5.45 Let Yi ,Fi , 1 ≤ i ≤ n, be a sequence of martingale differences. For anyA > 0, define Xi = Yi1

j

≤A

.

(i) Show that Xi ,Fi , 1 ≤ i ≤ n, is also a sequence of martingale differences. Here, the summation



j<1 Y 2 j is understood as zero.

(ii) Show that



n i=1 X2 i

≤ A + max1≤i≤n Y 2 i . (Hint: Define i

∗ = max{0 ≤

i ≤ n :



j

≤ A} and show that



n i=1 X2 i

≤



i

∗

i=1 Y 2 i .)

(iii) Show that



m i=1 Yi =



m i=1 Xi, 1 ≤ m ≤ n, on {



i

≤ A}.

(iv) Derive the following inequality. For any λ,A > 0 and p > 1, there is a constant c depending only on p such that P



max 1≤m≤n



m i=1 Yi



≥ λ,



i

≤ A



≤ c

λp Ap/2 + E



max 1≤i≤n

|Yi |p



.

[Hint: Use the results of (i)–(iii), (5.90) and Burkholder’s inequality.]