Abstract
We prove Wittmann’s SLLN (see Wittmann in Stat Probab Lett 3:131–133, 1985) for \(\psi \)-mixing sequences without the rate assumption.
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1 Introduction and result
Let \(\{X_k\} \), \(k=1,2,\ldots \) be a random sequence defined on a probability space \((\Omega , \mathcal{F}, P)\) and \(S_n=\sum _{k=1}^n X_k\). Set \({\mathcal{F}_{\!\!{k}}^{\,m}}\) the \(\sigma \)–field generated by \(X_k, X_{k+1},\ldots , X_m\), \(m\in {{\mathbb {N}}}\cup \{\infty \}\), and recall the following coefficients of dependence
Some of these measures of dependence can be expressed in language of norms (see Theorem 4.4 on p. 124, Vol. I [5]). In particular we have
We say that \(\{ X_k\}\) is \(\psi \)-mixing if \(\lim \limits _{n\rightarrow \infty }\psi _n=0\). It is well-known that (cf. Chapter 3 &5 in Volume I, [5])
See Remarks 5.23 on p. 186 in Vol. I, [5] for examples of mixing sequences ensuring that the relations in (1.2) are exact and Ch. 26 in Vol. III, [5] for possible mixing rates.
The famous random sequence satisfying \( \psi \)-mixing form digits of continued fraction expansion of irrational numbers. A lot of classical limit theorems for independent random variables hold for functionals of these digits (see [23]). However the rate of mixing in this case is exponential. On the other hand Kesten & O’Brien and Bradley gave examples of \( \psi \)-mixing sequences with arbitrary rate of mixing (see Ch. 3 and Ch. 26 in [5]). Therefore it is interesting to identify these classical limit theorems which mixing analogs do not require the rate assumption. In this paper we prove such strong results for sums and weighted sums of \( \psi \)-mixing sequences. Recall that non-parametric regression function estimators, empirical distribution functions and least squares estimators in statistics are weighted.
The following result is a generalization of Theorem 2.20 on p. 40 in [10] (see [3]). The new is the case when \(p>1\) and the proof which hangs on the Rosenthal maximal inequality (for \( \varrho \)-mixing case see [22], for blockwise m-NA case see [25]). For the case \(p\in ({{1}\over {2}},1)\) use the proof of Lemma 2.1 in [11].
Theorem 1.1
Suppose \(\{X_k\}\) is a \( \psi \)-mixing random sequence and \({ E}(X_k)=0 \), \(k\in {{\mathbb {Z}}}\). Let \(\{a_n\} \), \(a_0>0 \), be an increasing to infinity sequence of real numbers such that for some \(p\ge 1\)
-
(i)
\(\quad \sum _{n=1}^{\infty } a_n^{-2p}{ E}(|X_n|^{2p})<\infty ,\)
-
(ii)
\(\quad \sum _{n=1}^{\infty } a_n^{-2}(a_n^2-a_{n-1}^2)^{1-p}({ E}(X_n^2))^p < \infty \),
-
(iii)
\(\sup _n a_n^{-1}\sum _{k=1}^n E|X_k|<\infty . \)
Then, \(a_n^{-1}S_n\rightarrow 0 \), \(n\rightarrow \infty \), almost surely (a.s.).
Remark 1.2
It is clear from the proof of Theorem 1.1 that if \(\psi _m=0 \), for some \(m\ge 1\) then, the condition (iii) can be omitted because \(\{X_k\}\) is m-dependent.
By Theorem 1.1 we have the following variant of the Brunk–Prokhorov–Chung SLLN (see Corollary on p. 259 in [15](p. 271(Vol I) in 4th ed.) or Supplement 1 on pp. 286–7 in [20]).
Theorem 1.3
Suppose \(\{X_k\}\) is \( \psi \)-mixing, \({E}(X_k)=0 \), and for each \(k \ge 1\) \((a_k-a_{k-1})^{p-1}\ge C>0\), \(p\ge 1\). If \(\sup _n a_n^{-1} \sum \limits _{k=1}^n E|X_k|I(|X_k|\le a_k^{{{(p+1)}/{(2p)}}})<\infty \) and
then \(a_n^{-1}S_n\rightarrow 0\) as \(n\rightarrow \infty \) almost surely.
The proof of Theorem 1.1 yields a variant of Feller’s theorem (see [8, 20] pp. 274–275, [4, 7, 12, 13, 17, 19, 21]).
Theorem 1.4
Let \(\{X_k\}\) be a random sequence such that \({\mathcal L}(X_1)={{\mathcal {L}}}(X_k) \), \(k=2,3,\ldots \) and \(\{a_n\} \), be an increasing to infinity sequence of positive numbers and \(b_n>0\). Set \(c_n={{a_n}\over {b_n}}\). If
and
and \(\{X_k\}\) is \( \psi \)-mixing, then
Conversely, if (1.6) holds and either \(0<\psi '_m\) or \(\psi _m^{*}<\infty \) for some \(m\ge 1\) then, (1.5) holds, too.
Remark 1.5
If \(X_k\ge 0 \), \(E(X_1)=\infty ,\) \(L(x)=E(X_1I(X_1\le x))\) is slowly varying, in Theorem 1.4 we can take \(a_n\in {{\mathbb {R}}},\) \(a_n\uparrow \) such that \(a_n^{-1}\sum _{k=1}^n {{a_k}\over {kL^{\beta -1}(k)}}\sim 1 \), \(\beta >1\) and \(c_n = n L^{\beta }(n)\) to get \(a_n^{-1}\sum _{k=1}^n {{a_k}\over {kL^{\beta }(k)}} X_k \underset{\mathrm{a.s.}}{\rightarrow }1,\) \(n\rightarrow \infty \), if (1.5) holds.
As an application of Theorem 1.4 consider \(\{X_k\}\) with generalized St. Petersburg marginal, i.e. \(P(X_1 = q^{-k}) = pq^{k-1} \), \(0<p=1-q<1 \), \(k=1,2, \ldots \).
Proposition 1.6
Suppose that \(\{X_k\}\) is \( \psi \)-mixing then for \(\alpha >0\)
where \(\log \) is with respect to the base \(q^{-1}\).
Remark 1.7
In view of the proof of Theorem 1.4 and Theorem 6.11 on p. 181 in [18] Proposition 1.6 is true if \( \psi \)-mixing is replaced by \(\sum _{k=1}^{\infty } \varrho _{2^k}<\infty \). For other extensions see [2] and for the bibliography on the St. Petersburg Paradox up to the end of the previous millenium see [9].
2 Proofs
Proof of Theorem 1.1
Suppose first that \(\{X_k\}\) is a martingale difference such that \(E(X_k^2\,|\,\mathcal{F}_{k-1})\le (\psi _1+1)E(X_k^2)\) a.s. Let \(n_k\) be such that \(2a_{n_k}\le a_{n_{k+1}}\le 8a_{{n_k}+1}\) (see Lemma 2 in [24]). By Markov’s inequality and Theorem 2.11 in [10] we have a.s. (\(n_0=0\))
Thus by the proof on p. 132 in [24] we get \(\lim \limits _{n\rightarrow \infty }a_n^{-1}S_n = 0\) a.s.
For general case fix \(\epsilon \ge 0\) and m such that \(\psi _m\le \epsilon \), \(n=qm+r \), \(0\le r<m\). Decompose
(see [3] and p.40 in [10]). Clearly,
as \(n\rightarrow \infty \) since \(r<m\) and m is fixed. By (1.1)
Therefore,
Fix i and set \(Y_{nk}(i)=X_{km+i} - E(X_{km+i}\vert X_{i},X_{2\,m+i}, \ldots , X_{(k-1)m+i})\) and \(Z_n(i)=\sum _{k=1}^{q-1} Y_{nk}(i)\). Since \(Y_{nk}(i)\) are martingale difference and \(a^2_{km+i} - a_{km+i-1}^2 \le a^2_{km+i} - a_{(k-1)m+i}^2\) by the previous step \(\lim \limits _{n\rightarrow \infty }a_{(q-1)m+i}^{-1}Z_n(i) = 0\) a.s. for each i separately, and we obtain by the Kronecker lemma (see Theorem 3 on p. 129 in [14])
By (2.5), (2.4) and (2.2) Theorem 1.1 follows. \(\square \)
Poof of Theorem 1.3
We apply Theorem 1.1 for \({{\bar{X}}}_k=X_kI(|X_k|\le a_k^{{ {(p+1)} / {(2p)} }})-EX_kI(|X_k|\le a_k^{{ {(p+1)} / {(2p)} }})\). We have
thus (iii) holds. Further, in view of
(ii) and (i) hold by (1.3). Finally, by the Markov inequality,
Since \(E(X_k)=0\),
as \(n\rightarrow \infty \), by the Kronecker lemma. Theorem 1.3 is proved. \(\square \)
Remark 2.1
Let \(x_n,y_n\) be a nonegative real sequences such that \(y_n\nearrow \infty \). If we take \(a_{n\nu }={{y_{\nu }-y_{\nu -1}}\over {y_n}}\) and \(s_{\nu }={{x_{\nu }-x_{\nu -1}}\over {y_{\nu }-y_{\nu -1}}}\) in Theorem 1.3 on p. 75 in [26] then, we obtain the following generalization of the Stolz theorem
Since \((a_k-a_{k-1})^{p-1}\ge C>0\), \(p\ge 1,\) we can by (2.6) replace in Theorem 1.3 the condition
with \(\limsup \limits _{n\rightarrow \infty } E|X_n|I(|X_n|\le a_n^{{ {p+1} / {(2p)} }})<\infty \).
Proof of Theorem 1.4
For the direct statement we have
where the last equality follows by summation by parts (see p.422 in [16]). Having this we use Theorem 1.1 with \(p=1\) for \({{\bar{X}}}_k=b_k(X_kI(|X_k|\le c_k)-EX_kI(|X_k|\le c_k))\) and \(a_n\). Finally, by (1.5) and the first Borel–Cantelli lemma
so the direct half holds.
For the converse statement write \({{\tilde{S}}}_{mk} = \sum _{i=1}^{mk} b_iX_iI(|X_i|>c_i) + {{\bar{X}}}_i \), \(k=1,2,\ldots \) We have
since \(\lim \limits _{k\rightarrow \infty }a_{mk}^{-1}b_{mk}EX_{mk}I(|X_{mk}|\le c_{mk}) = 0\). Therefore \(P(A_k \> i.o.)=0\), where \(A_k=\{|X_{mk}|>c_{mk}\}\). We have
so if \(\sum _{k=1}^{\infty } P(A_k)\) diverges then
Therefore,
Whence the Rényi–Lamperti lemma (see Theorem 3.2.1 on p. 66 in [6]) yields \(P(A_k\>\, {i.o.})\ge {{1}\over {\psi _m^{*}}}>0\). Since we get a contradiction \(\sum _{k=1}^{\infty } P(A_k)<\infty \). For the case \(\psi '_m>0\) use Theorem 3.6.1 on p. 78 in [6]. Thus we have
and (1.5) holds. Theorem 1.4 is proved. \(\square \)
Remark 2.2
In the converse statement we need in fact weaker conditions, namely \(P(A_{k}\cap A_{j})\ge (\le ) {\eta }' ({\eta }^*) P(A_{k})P(A_{j})\) for some \({\eta }' ({\eta }^*)\in (0,\infty )\) and for each \(k\ne j\).
Proof of Proposition 1.6
Take in Theorem 1.4\(a_n=\log ^{\alpha }{n} \), \(b_n={{(\log {n})^{\alpha -2}}\over {n}}\) so that \(c_n=n\log ^2{n}\). By the Stolz theorem for null sequences (see Ex.29 on p. 109 in [14]) we get
by the regular variation of \(n^{-1}c_n^2\). We have \(E(X_1I(X_1\le x))\sim {{p}\over {q}}\log {x}\) and \(xP(X_1>x)\le q^{-1}\) (see [1]). Therefore,
by Cauchy’s condensation test (see Theorem 3.27 on p. 208 in [16]) and
as \(n\rightarrow \infty \), (see Theorem 8.6 on p. 15 in [26]). \(\square \)
References
A. Adler, Exact strong laws. Bull. Inst. Math. Sin. 28(5), 141–146 (2000)
A. Adler, P. Matuła, On exact strong laws of large numbers under general dependence conditions. Probab. Math. Statist. 38(1), 103–121 (2018)
J.R. Blum, D.L. Hanson, L.H. Koopmans, On the Strong Law of Large Numbers for a Class of Stochastic Processes. Z. Wahr. verw. Gebiete 2, 1–11 (1963)
F. Boukhari, On a Feller-Jajte strong law of large numbers. Comm. Statist. Theory Methods 51(18), 6218–6226 (2022)
R.C. Bradley, Introduction to Strong Mixing Conditions, vol. I–III (Kendrick Press, Heber City, 2007)
T.K. Chandra, The Borel-Cantelli lemma (Springer, Cham, 2012)
P. Chen, S.H. Sung, On the Jajte strong law of large numbers. Statist. Probab. Lett. 176, 109138 (2021)
W. Feller, A limit theorem for random variables with infinite moments. Amer. J. Math. 68(2), 257–262 (1946)
S. Csörgő, G. Simons, A bibliography of the St. Petersburg Paradox (Hungarian Academy of Sciences, Szeged) (2000)
P. Hall, C.C. Heyde, Martingale Limit Theory and its Applications (Academic Press, New York, 1980)
D. Hu, P. Chen, S.H. Sung, Strong laws of weighted sums of \( \psi \)-mixing random variables and applications in errors-in-variables regression models. TEST 26, 600–617 (2017)
R. Jajte, On the Strong Law of Large Numbers. Ann. Probab. 31(1), 409–412 (2003)
B.-Y. Jing, H.-Y. Liang, Strong limit theorems for weighted sums of negateively associated random variables. J. Teor. Probab. 21, 890–909 (2008)
K. Knopp, Theory and Application of Infinite Series (Blackie &Son Ltd., London-Glasgow, 1954)
M. Loève, Probability Theory, 2nd edn. (Van Nostrand, New York, 1960)
P. Loya, Amazing and Aesthetic Aspects of Analysis (Springer, Cham, 2017)
A.I. Martikainen, V.V. Petrov, On a Theorem of Feller. Theory Probab. Appl. 25(1), 191–193 (1980)
F. Merlevède, M. Peligrad, S. Utev, Functional Gaussian Approximation for Depenedent Structures (OUP, Oxford, 2019)
Y. Miao, J. Mu, S. Zhang, Limit theorems for identicall distributed martingale differences. Comm. Statist. Theory Methods 49(6), 1435–1445 (2020)
V.V. Petrov, Sums of Independent Random Variables (Springer, Cham, 1975)
M. Seweryn, Almost sure limit theorems for dependent random variables. Banach Center Publ. 90, 171–178 (2010)
Z.S. Szewczak, A moment maximal inequality for dependent random variables. Statist. Probab. Lett. 106, 129–133 (2015)
Z.S. Szewczak, On Kolmogorov’s converse inequality for dependent random variables. Stoch. Anal. Appl. 39(3), 483–493 (2021)
R. Wittmann, An application of Rosenthal’s moment inequality to the Strong Law of Large Numbers. Statist. Probab. Lett. 3, 131–133 (1985)
G. Zhang, H. Urmeneta, A. Volodin, Wittmann type strong laws of large numbers for blockwise m-negatively associated random variables. J. Math. Res. Appl. 36(2), 239–246 (2016)
A. Zygmund, Trigonometric Series (volumes I & II Combined), 3rd edn. (Cambridge University Press, Cambridge, 2002)
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The author thanks Péter Kevei and the anonymous reviewer for suggestions that improved the presentation.
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Szewczak, Z.S. On the Wittmann strong law for mixing sequences. Period Math Hung (2024). https://doi.org/10.1007/s10998-024-00588-z
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DOI: https://doi.org/10.1007/s10998-024-00588-z
Keywords
- Strong laws of large numbers (SLLN)
- \(\psi \)
- \(\psi ^{*}\)
- \(\psi '\) coefficients of depedence
- St. Petersburg game