# Download e-book for kindle: Almost Sure Convergence by William F. Stout By William F. Stout

ISBN-10: 0126727503

ISBN-13: 9780126727500

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Extra resources for Almost Sure Convergence

Sample text

1) yields f Jb < ^= dP + f J BC\(t>n) u: - J B n (t> n ) dP «J B n (t < n ) dP + f «1 B n (t< n ) t/»-i dP = f U£x dP, J B establishing the theorem. R e m a rk s. s. However, even if t is almost surely finite it is not necessarily true that EU1 = EUt. This misunder­ standing has sometimes led to serious errors. As a simple example, let {Yi, i > 1} be independent identically distributed with P[YX = 1] = PiY^ = —1] = \ (coin tossing model). Let Un = Ya=i Yt for each n > 1 define the martingale {Un, n > 1}.

There are two cases: either 1 < n < N/2 or N/2 < n < N. If 1 < n < N/2, / a+n \2 ( I \i= a + l / If N/2 < n < N, i a+n \2 /a+N/2 u = a+ 1 . E *) = ( Z . * + . x /a+A72 * i=*a+A72+l \2 \! / /a+xV/2 \, a+n a+n . l **)(.. / \i»a+ A 72+ lX i) + . 4 25 CO NVERGENCE ASSUMING O N LY MOMENT RESTRICTIONS Taking expectations on both sides and applying the induction hypotheses, we obtain a+N/2 EM2a,N < (log N1log 2)2g(FafN/2) + 2E\ . E '¿=0+1 +(log JV/log 2)2g(Fa+N/2'N/2). 4) \ 2 ^ E ' > * № +NI2'NI2) \i=a+l / < 2(log JV/log 2)g"2(Fa,*/2)g"2(Fa+ W /2 ) < (log A'/log 2)tg(Fa Ar/2) + g(>a+W2tAr/2)], using the Cauchy-Schwarz inequality, Eq.

This fact is helpful in 36 POINTWISE CO N VERGEN CE OF PARTIAL SUMS Chapter 2 studying the behavior of Tn. For further discussion, see the work of Doob [1953, p. 348]. 4. s. s. | Note that a martingale difference sequence is trivially a submartingale difference sequence and that a martingale is trivially a submartingale. Note also that {Tn, re > 1} defined by Tn = £ ”=1 for each re > 1 is a sub­ martingale if and only if {Y i, i > 1} is a submartingale difference se­ quence. If {Y i, i > 1} is a positive sequence of random variables (that is, P[Yi 0] — 1 for each i 1) then, trivially, [ , i '^> 11 is a submartingale difference sequence.