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Convergence of random power series with pairwise independent Banach-space-valued coefficients
Statistics & Probability North-Holland
Letters
18 (1993) 121-123
22 September
1993
Convergence of random power series with pairwise independent Banach-space-valued coefficients Markus Universitiit
Roters Trier, Germany
Received November 1991 Revised January 1993
Abstract: The distribution of the radius of convergence of a random power series with pairwise independent and non-identically distributed Banach-space-valued coefficients is considered. The results obtained here extend the well-known work in the complex-valued, identically distributed case and provide a correction of a theorem in Rohatgi (1975) concerning independent, non-identically distributed coefficients. Keywords:
Random
power
series; radius
of convergence;
zero-one
1. Introduction
cc
W) =
Borel-Cantelli
lemma.
a n,
Let (0,9, P) be a probability space and (a,), E N, a sequence of random elements with values in a real or complex Banach space (B, II. II>, B z {O). Considering the random power series f(z,
law; refined
C a,(0)zn,
z E R or @,
n=O
it is well-known that the radius of convergence r(o) E LO, ~1 of f< z, w) is given by the formula of Cauchy-Hadamard, i.e., r(w)=l/limsupIIa,(w)II”” n+oo (cf. Conway, 1985, proof of Proposition VII, 3.8). In the case of independent random variables
12E N,, Kolmogorov’s zero-one law implies that r is a.s. constant. This case is treated, for example, in Arnold (1966) or Lukacs (1975). In the dependent case a paper by Holgate (1970) investigates exchangeable processes, stationary time series or denumerable Markov chains in equilibrium as random coefficients a,,, n E N,. Moreover, a paper by Rohatgi (1975) contains as its main result (Theorem 3) a characterization of the radius of convergence for independent, but not necessarily identically distributed coefficients, which is false in its present formulation. The aim of this paper is to present a counterexample to Rohatgi’s theorem and to prove a correct version of it, stated below as Theorem 2*1.
2. The main result Correspondence to: Statistik, Universitlt many. 0167-7152/93/$06.00
Markus Roters, Trier, Postfach
FB IV Mathematik/ 3825, 5500 Trier, Ger-
0 1993 - El sevier Science
Publishers
The following two conditions will play an important role in the sequel.
B.V. All rights reserved
121
Volume
18, Number
STATISTICS&PROBABILITY
2
(A) There exists a B-valued random element a defined on (0, 9, P) with P(a # 0) > 0 such that for every n E N, and x E 10, co), P(Il%J
>x)>~(lbll
>x>
is fulfilled. (B) There exists a B-valued random element b defined on (0, F’, P) with P(b # 0) > 0 such that for every n E Ni, and x E [O, m),
P(lta,II
>x)
Ilbll
>x)
is satisfied. Theorem 3 is Rohatgi (1975) states that under the assumption of (B) for independent, complexvalued coefficients a,, n E N,, the radius of convergence r = 1 or 0 a.s. according as E(log+ Ilb II> <~or=w. However, this theorem turns out to be incorrect as can be seen from a counterexample given below. In this paper we prove a corrected and generalized version of Rohatgi’s theorem by only using the first and the refined second BorelCantelli lemma (cf. Billingsley, 1986, p. 84) instead of the three-series criterion of Kolmogorov utilized in Rohatgi’s work. Now, our main result is the following: Theorem 2.1. Let Cz=, a,z” be a random power series with B-valued coefficients a,,, n E N,. Suppose that the IIa,, II, n E N,, are pairwise independent. Then it follows: (i) If (A), (B) and E(log’ IIb II>< 00 are fulfilled, then r = 1 a.s. (ii) If (A) and E(log+ 11a 11)= 00 are satisfied, then r = 0 a.s. Here, log+= max(O, log). Remark together If an random = 0 and This Theorem sumption.
122
2.2. Theorems 1 and 2 in Rohatgi (1975) generalize to: arbitrary sequence (a,), E NJ, of B-valued elements satisfies (A), (B), P( IIa II = 0) E(log+ 11b 11)< 00 then r = 1 a.s. is obviously the same implication as in 2.1(i) under somewhat different as-
22 September
LETTERS
1993
Before beginning the proof, we state a lemma generalizing Kolmogorov’s zero-one law for the special tail function r = l/ lim sup 11a, 111’n. Lemma 2.3. Under the assumptions of Theorem 2.1 the radius of convergence r is an a.s. constant non-negative extended real-valued random variable. Proof. Assume P(r < w) > 0, then there k E (0, UJ>with P(r 0 and hence lim sup { II a, IIl” > l/k}) n+m
> 0.
But in this case, the first and the refined Borel-Cantelli lemma imply P lim sup ( IIa, I11’n > l/k}) ( n-m
exists a
second
= 1,
thus P( r G k) = 1. Considering k, = min( k E [0, m) I P( r < k) = 1) < m, we obtain P(r G k,) = 1 and P(r < k,) = 0, hence P(r = k,) = 1 is satisfied. But his proves Lemma 2.3. •I Proof of Theorem 2.1. (i) In view of the assumptions (B) and E(log+ II b II>< CCthe same simple Borel-Cantelli argument as in the proof of Theorem 2 in Rohatgi (1975) yields r > 1 a.s. Moreover, from the pairwise independence of the 11a, 11, n E N,, Lemma 2.3 entails that r is a.s. constant. Finally, under the further assumption (A), and hence E(log+ IIa II> < m, the same proof as Theorem 1 of Rohatgi (1975) yields P(r > 1) < 1, which consequently implies the assertion of (i). (ii) To prove this part notice that E(log+ Ila II> = ~0 yields 5
n=O
P( II a, II > q
for all 6 > 1. Hence, lemma,
2 2 P( II u II > 6”) = cc n=O by the refined
P lim sup { II a, II1’n > s}) = 1, n+m
Borel-Cantelli
Volume
18, Number
2
STATISTICS&PROBABILITY
= P( lim sup IIa, II1’n > 8) rZ+‘= 2 P lim sup { II a, II1/n > s}) = 1, ( n--rm
so that
r = 0 a.s. This proves
the theorem.
1993
(1975) are fulfilled, in case (il it is E(logCb)) = 0 < to, whereas in case (ii) E(log(b)l = 03 holds, but obviously
and consequently P( r < l/6)
22 September
LETTERS
r = l/lim
supa:‘” rl+m
in contradiction
=p E {O,l} to Rohatgi’s
theorem.
q
Corollary 2.4. Zf ( IIa,, II),,E N,, is a sequence of identically distributed, pairwise independent random variables with P(a, # 0) > 0 the assumptions of Theorem 2.1 are trivially fulfilled, hence (i) E(log+ IIa, II>< cQiff r = 1 a.s. (ii) E(log+ 11a, 11)= m iff r = 0 a.s. This is the zero-one law for random power series in the i.i.d. case (cf. Arnold, 1966, Satz 6). 0 Counterexample 2.5. Let p E (1, w) and a,, = l/p” for all n E N,. If(i) b = 1 or (ii) b is a real-valued random variable with P(b = 22”> = l/2” for all n E N, the assumptions of Theorem 3 in Rohatgi
References Arnold, L. (1966), uber die Konvergenz einer zufalligen Potenzreihe, J. Reine Angew. Math. 222, 79-112. Billingsley, P. (19861, Probability and Measure (Wiley, New York). Conway, J.B. (1985), A Course in FunctionalAnalysis(Springer, New York). Holgate, P. (1970), Power series whose coefficients form homogeneous random processes, Z. Wahrsch. Vera. Gebiete 15, 97-103. Lukacs, E. (19751, Stochastic Convergence (Academic Press, New York, 2nd ed.). Rohatgi, V.K. (1975), On random power series with nonidentically distributed coefficients, J. Multiuariate Anal. 5, 265270.
123
Letters
18 (1993) 121-123
22 September
1993
Convergence of random power series with pairwise independent Banach-space-valued coefficients Markus Universitiit
Roters Trier, Germany
Received November 1991 Revised January 1993
Abstract: The distribution of the radius of convergence of a random power series with pairwise independent and non-identically distributed Banach-space-valued coefficients is considered. The results obtained here extend the well-known work in the complex-valued, identically distributed case and provide a correction of a theorem in Rohatgi (1975) concerning independent, non-identically distributed coefficients. Keywords:
Random
power
series; radius
of convergence;
zero-one
1. Introduction
cc
W) =
Borel-Cantelli
lemma.
a n,
Let (0,9, P) be a probability space and (a,), E N, a sequence of random elements with values in a real or complex Banach space (B, II. II>, B z {O). Considering the random power series f(z,
law; refined
C a,(0)zn,
z E R or @,
n=O
it is well-known that the radius of convergence r(o) E LO, ~1 of f< z, w) is given by the formula of Cauchy-Hadamard, i.e., r(w)=l/limsupIIa,(w)II”” n+oo (cf. Conway, 1985, proof of Proposition VII, 3.8). In the case of independent random variables
12E N,, Kolmogorov’s zero-one law implies that r is a.s. constant. This case is treated, for example, in Arnold (1966) or Lukacs (1975). In the dependent case a paper by Holgate (1970) investigates exchangeable processes, stationary time series or denumerable Markov chains in equilibrium as random coefficients a,,, n E N,. Moreover, a paper by Rohatgi (1975) contains as its main result (Theorem 3) a characterization of the radius of convergence for independent, but not necessarily identically distributed coefficients, which is false in its present formulation. The aim of this paper is to present a counterexample to Rohatgi’s theorem and to prove a correct version of it, stated below as Theorem 2*1.
2. The main result Correspondence to: Statistik, Universitlt many. 0167-7152/93/$06.00
Markus Roters, Trier, Postfach
FB IV Mathematik/ 3825, 5500 Trier, Ger-
0 1993 - El sevier Science
Publishers
The following two conditions will play an important role in the sequel.
B.V. All rights reserved
121
Volume
18, Number
STATISTICS&PROBABILITY
2
(A) There exists a B-valued random element a defined on (0, 9, P) with P(a # 0) > 0 such that for every n E N, and x E 10, co), P(Il%J
>x)>~(lbll
>x>
is fulfilled. (B) There exists a B-valued random element b defined on (0, F’, P) with P(b # 0) > 0 such that for every n E Ni, and x E [O, m),
P(lta,II
>x)
Ilbll
>x)
is satisfied. Theorem 3 is Rohatgi (1975) states that under the assumption of (B) for independent, complexvalued coefficients a,, n E N,, the radius of convergence r = 1 or 0 a.s. according as E(log+ Ilb II> <~or=w. However, this theorem turns out to be incorrect as can be seen from a counterexample given below. In this paper we prove a corrected and generalized version of Rohatgi’s theorem by only using the first and the refined second BorelCantelli lemma (cf. Billingsley, 1986, p. 84) instead of the three-series criterion of Kolmogorov utilized in Rohatgi’s work. Now, our main result is the following: Theorem 2.1. Let Cz=, a,z” be a random power series with B-valued coefficients a,,, n E N,. Suppose that the IIa,, II, n E N,, are pairwise independent. Then it follows: (i) If (A), (B) and E(log’ IIb II>< 00 are fulfilled, then r = 1 a.s. (ii) If (A) and E(log+ 11a 11)= 00 are satisfied, then r = 0 a.s. Here, log+= max(O, log). Remark together If an random = 0 and This Theorem sumption.
122
2.2. Theorems 1 and 2 in Rohatgi (1975) generalize to: arbitrary sequence (a,), E NJ, of B-valued elements satisfies (A), (B), P( IIa II = 0) E(log+ 11b 11)< 00 then r = 1 a.s. is obviously the same implication as in 2.1(i) under somewhat different as-
22 September
LETTERS
1993
Before beginning the proof, we state a lemma generalizing Kolmogorov’s zero-one law for the special tail function r = l/ lim sup 11a, 111’n. Lemma 2.3. Under the assumptions of Theorem 2.1 the radius of convergence r is an a.s. constant non-negative extended real-valued random variable. Proof. Assume P(r < w) > 0, then there k E (0, UJ>with P(r
> 0.
But in this case, the first and the refined Borel-Cantelli lemma imply P lim sup ( IIa, I11’n > l/k}) ( n-m
exists a
second
= 1,
thus P( r G k) = 1. Considering k, = min( k E [0, m) I P( r < k) = 1) < m, we obtain P(r G k,) = 1 and P(r < k,) = 0, hence P(r = k,) = 1 is satisfied. But his proves Lemma 2.3. •I Proof of Theorem 2.1. (i) In view of the assumptions (B) and E(log+ II b II>< CCthe same simple Borel-Cantelli argument as in the proof of Theorem 2 in Rohatgi (1975) yields r > 1 a.s. Moreover, from the pairwise independence of the 11a, 11, n E N,, Lemma 2.3 entails that r is a.s. constant. Finally, under the further assumption (A), and hence E(log+ IIa II> < m, the same proof as Theorem 1 of Rohatgi (1975) yields P(r > 1) < 1, which consequently implies the assertion of (i). (ii) To prove this part notice that E(log+ Ila II> = ~0 yields 5
n=O
P( II a, II > q
for all 6 > 1. Hence, lemma,
2 2 P( II u II > 6”) = cc n=O by the refined
P lim sup { II a, II1’n > s}) = 1, n+m
Borel-Cantelli
Volume
18, Number
2
STATISTICS&PROBABILITY
= P( lim sup IIa, II1’n > 8) rZ+‘= 2 P lim sup { II a, II1/n > s}) = 1, ( n--rm
so that
r = 0 a.s. This proves
the theorem.
1993
(1975) are fulfilled, in case (il it is E(logCb)) = 0 < to, whereas in case (ii) E(log(b)l = 03 holds, but obviously
and consequently P( r < l/6)
22 September
LETTERS
r = l/lim
supa:‘” rl+m
in contradiction
=p E {O,l} to Rohatgi’s
theorem.
q
Corollary 2.4. Zf ( IIa,, II),,E N,, is a sequence of identically distributed, pairwise independent random variables with P(a, # 0) > 0 the assumptions of Theorem 2.1 are trivially fulfilled, hence (i) E(log+ IIa, II>< cQiff r = 1 a.s. (ii) E(log+ 11a, 11)= m iff r = 0 a.s. This is the zero-one law for random power series in the i.i.d. case (cf. Arnold, 1966, Satz 6). 0 Counterexample 2.5. Let p E (1, w) and a,, = l/p” for all n E N,. If(i) b = 1 or (ii) b is a real-valued random variable with P(b = 22”> = l/2” for all n E N, the assumptions of Theorem 3 in Rohatgi
References Arnold, L. (1966), uber die Konvergenz einer zufalligen Potenzreihe, J. Reine Angew. Math. 222, 79-112. Billingsley, P. (19861, Probability and Measure (Wiley, New York). Conway, J.B. (1985), A Course in FunctionalAnalysis(Springer, New York). Holgate, P. (1970), Power series whose coefficients form homogeneous random processes, Z. Wahrsch. Vera. Gebiete 15, 97-103. Lukacs, E. (19751, Stochastic Convergence (Academic Press, New York, 2nd ed.). Rohatgi, V.K. (1975), On random power series with nonidentically distributed coefficients, J. Multiuariate Anal. 5, 265270.
123