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Stoyanov J. Counterexamples in Probability (2ed., Wiley, 1997)(T)(366s).djvu |
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AB is finite and 0 ^ A, or Ac?? is finite and 0 ^ Ac...
Denote by Pq and Pn the
measures on E induced by Fo and Fn and let Q be the set of all rational numbers in
E1...
Consider
the d.f.s F and Fn, n = 1,2,..., defined by
Г 0 f < 0 f ^' if x < -n
F(x) — < ' ....
Take another
sequence {vn,n > 1} of integer-valued r.v.s such that vn —4 oo as n —> oo and
define Tn = Svn = Xi + ¦ ¦ ¦ + XVn and 62 = VTn...
For this let {ап(оо), п > 0} be independent
r.v.s with
P[a0 = 0] = 1, P[an = nn] - l/n, P[an = 0] = 1 - l/n, n > 1...
n—юо
We write U G N(-y) to denote that U is in the domain of normal attraction of a
stable law with index 7...
From C) and D) it follows that
E)
Taking B) into account we find that
00
k=2
204 COUNTEREXAMPLES IN PROBABILITY
On the other hand, since logn^+i ~ k/(C\ogk), for any fixed S, 0 < S < |, we
obtain from E) that
oo
k=2
However, these two series must converge or diverge together...
Therefore
P[ lim т]п =
n—Юо
Applying a result of Zolotarev and Korolyuk A961) we see that the sequence {?n}
cannot converge weakly (to a non-degenerate r.v.) for any choice of the norming
constants {bn}...
However, if t\ ф t2, t\, t2 G T, the
events [Xtt = 1] and [Xtl = 1] cannot be equivalent, since this would give
\ = Y[Xtx = \) = P[Xtl = \,Xt2 = 1] ф P[Xtl = \)P[Xt2 = 1] = J...
The independence
between Xt and Xs, even for very close s and t, is inconsistent with the desired
continuity of the process...
Clearly, if the process X is separable, then any event associated with X can
be represented by countably many operations like union and intersection...
JnJi Ji
Hence using the Fubini theorem we obtain
Xt(w)dt) >=e| Г f Xs{u)Xt{uj)dsdt\= f IE[XsXt]dsdt = 0...
On the stochastic continuity and the weak L1 -continuity of stochastic
processes
Let X = (X(t), t ? T) be a stochastic process where T is an interval in IR1...
Thus X{0) = 0, X{\/n) = rjn, n > 1, and
for all w€ii, X(-,u;) is a linear function on every interval [^j, ^], ^ > 1...
Important general results concerning continuity properties of stochastic processes,
including some useful counterexamples, are given by Ibragimov A983) and
Balasanov and Zhurbenko A985)...
Then it is not difficult to find that
where 7(s, t) is exactly the function introduced above...
Now take again the above one-point probability space (Q, 1, P) and choose any
function h : E+ »-> E1...
Some important
books are devoted to the general theory of Markov processes: see Dynkin A961,
1965), Blumenthal and Getoor A968), Rosenblatt A971, 1974), Wentzell A981),
Chung A982), Ethier and Kurtz A986), Bhattacharya and Waymire A990) and
Rogers and Williams A994)...
Consider now the new process {Yn,n = 0,1,2,...} where Yn = g(Xn) and g
is a given function on E...
Firstly, let us note that the transition probabilities of any continuous-time Markov
chain satisfy two systems of differential equations which are called Kolmogorov
equations (see Chung 1960; Gihman and Skorohod 1974/1979)...
Indeed, by the formula
P(t,x,T) = Ir{x + vt), t >0, x G R1, Ге Ъ\ v = constant > 0
we define a transition function which corresponds to a homogeneous Markov process...
In this section we consider only a few examples dealing with the stationarity
property, as well as properties such as mixing and ergodicity...
Let g(x), x € Ш1, be a measurable function and ? = (?n,rc 6 N) a strictly
stationary process...
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