LETTERS
TO
THE
EDITOR
215
Theorem.
The
following
three
assertions
are
equivalent.
(1)
The
collection
Tk;i}
forms
a
basis
of
decomposition
which
are
e
mod
0.
(2)
The
Scode
A
{Ai}
covers
the
Bernoulli
scheme
(Xz,
Izp,
T),
or
does
not
cover
it,
but
the
alphabet
Z
contains
a
letter
ak
such
that
almost
every
realization
can
be
represented
(uniquely
up
to
an
indexing
]l,
a
possible
shift)
in
the
form
akBi_.ak
akBi_..,ak
akBjoak
Bj,
akBn,ak
"’’,
where
Bi,
o<
<
o,
is
a
word
of
any
kind;
i.e.
all
"lacunae"
between
words
of
whatever
kind
(for
some
there
may
be
none)
are
filled
by
letters
ak
in
arbitrary
number.
(3)
For
all
letters
aj
Z,
except
possibly
one
(the
ak
in
2)),
P(ai)
ktloA’l’lzp(.,),
where
to
is
the
smallest
positive
real
zero
of
the
polynomial
(series)
](t)
l
+
Zlzo(Ai)t
la’l.
The
equality
to
(1
Y./zp
(/i))
1
holds,
in
which
the
summation
ranges
over
all
words
of
any
kind.
There
is
coverability
when
all
equalities
in
(3)
are
satisfied
for
all
a..
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 Fall '11
 EugeneA.Feinberg
 Bernoulli, Markov chain, A. M. Vershik, problem jor Bernoulli, positive real zero, multiple positive root

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