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Feller W. An introduction to probability theory and its applications Vol II (3ed., Wiley, 1971)(T)(683s)_MV_.djvu |
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of the partial sums must converge, and so (9.3) and (9.4) hold...
It follows easily thaf ?}(/„, tn+1) ~* 1 and this implies
&('n> sn) — 1 for any sequence of epochs such that /„ < sn < tn+v In view of (9.5) this
means that Q(t, sn) -* U, and so the limit U is independent of the sequence {/„}, and
the lemma is proved...
If ju0 is the probability distribution at epoch 0 the distribution at epoch
t is given by
A,7)
and A.6) implies
A.8) B-^JP- = -m{F} + aj^{dz} K(z,
This version of A.6) is known to physicists as the Fokker-Planck (or
continuity) equation...
On the other hand, the backward equation is a necessary consequence of
the basic assumptions, and it is therefore best to use it as a starting point
and to investigate the extent to which the forward equation can be derived
from it...
We
shall therefore be satisfied with a derivation of the backward equation and
with a brief summary concerning the minimal solution and other problems...
MARKOV PROCESSES AND SEMI-GROUPS X.4
and the empirical meaning of the diffusion equations in the simplest situation...
It follows that
the transition densities qt(z, y) of the Ornstein-Uhlenbeck process coincide
with the normal density centered at ze~pt and with variance t given by D.1-2)...
In the preceding section we were able by probabilistic arguments to show
that, the transform D.5) satisfies the backward equation D.6)...
The catch is that for fixed
/ and x the kernel Ot(x,Y) may represent a defective distribution...
This solution can be obtained also by a routine application of the method of
Fourier series in the form20
E-9) ^-i
We nave thus obtained two very different representations21 for the same
function Aa...
>
22 The topic is relatively new, and yet the starting point of the much used identity E.W)
seems already to have fallen into oblivion...
This result may be interpreted in terms of two independent Brownian
motions X(/) and Y(/) as follows...
As in chapter VIII the norm of a bounded real function u is defined by
||i/1| = sup \u(x)\...
In this case we write
T -+¦ T
From now on we concentrate on semi-groups of contraction operators
and impose a regularity condition on them...
This operator being an endomorphism,
= v is defined for all u e S?, and
A0.1) Qih) * u -» v,
h
It would be pleasant if the same were true of all semi-groups, but this is
too much to expect...
In view of this impressive history a new and greatly simplified proof of the
general theorem is incorporated in section 9...
[With the convention that z(x) = Z{x) = 0 for x < 0 we may write
A.5) in the form Z = ?/**.]
Proof...
o 2/u Jo Ifx
Being monotone z is directly integrable, and the alternative form of the
renewal theorem asserts that C.1) is true...
(b) The density u of the preceding example satisfies C.6) which agrees with the standard
renewal equation A.3) with z = /...
Indeed, when /u == oo we have
D.11) H(t, ?)-*0
for all ?: the probability tends to 1 that the level t will be overshot by an
arbitrarily large amount ?...
Fcr the length Lt = SN(+1 — SN<
of the interarrival time containing the epoch t we get
D.15) P{L, < |} = f U{dz}[F($) - F(t-x)]
Jt-t
and hence
D.16) lim P{L< ? ?} = [T1 \\f& - F(y)] dy = /*"* [x F(dx)...
Examples, (a) The event {M <; t) occurs if the process terminates with
So, or else if Tx assumes some positive value y < t and the residual
process attains an age <, t — y...
[The process is
meaningful only if LK < 1 for otherwise R(i) = 0 for all t.\ Note that
G.2) is a special case of F.4) and that i?(oo) = 1...
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