Abstract
Let \((X_n)_{n\geqslant 0}\) be a Markov chain with values in a finite state space \({\mathbb {X}}\) starting at \(X_0=x \in {\mathbb {X}}\) and let f be a real function defined on \({\mathbb {X}}\). Set \(S_n=\sum _{k=1}^{n} f(X_k)\), \(n\geqslant 1\). For any \(y \in {\mathbb {R}}\) denote by \(\tau _y\) the first time when \(y+S_n\) becomes non-positive. We study the asymptotic behaviour of the probability \({\mathbb {P}}_x \left( y+S_{n} \in [z,z+a],\, \tau _y > n \right) \) as \(n\rightarrow +\infty .\) We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalent of order \(n^{3/2}\) and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability \({\mathbb {P}}_x \left( \tau _y = n \right) \) as \(n\rightarrow +\infty \).
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References
Bertoin, J., Doney, R.A.: On conditioning a random walk to stay nonnegative. Ann. Probab. 22(4), 2152–2167 (1994)
Bolthausen, E.: On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab. 4(3), 480–485 (1976)
Borovkov, A.A.: On the asymptotic behavior of distributions of first-passage times, I. Math. Notes 75(1–2), 23–37 (2004)
Borovkov, A.A.: On the asymptotic behavior of distributions of first-passage times, II. Math. Notes 75(3–4), 322–330 (2004)
Caravenna, F.: A local limit theorem for random walks conditioned to stay positive. Probab. Theory Relat. Fields 133(4), 508–530 (2005)
Denisov, D., Wachtel, V.: Conditional limit theorems for ordered random walks. Electron. J. Probab. 15, 292–322 (2010)
Denisov, D., Wachtel, V.: Exit times for integrated random walks. Ann. Inst. Henri Poincaré Probab. Stat. 51(1), 167–193 (2015)
Denisov, D., Wachtel, V.: Random walks in cones. Ann. Probab. 43(3), 992–1044 (2015)
Doeblin, W., Fortet, R.: Sur les chaìnes à liaisons complètes. Bull. Soc. Math. France 65, 132–148 (1937)
Doney, R.A.: On the asymptotic behaviour of first passage times for transient random walk. Probab. Theory Relat. Fields 81(2), 239–246 (1989)
Eichelsbacher, P., König, W.: Ordered random walks. Electron. J. Probab. 13, 1307–1336 (2008)
Gnedenko, B.V.: On a local limit theorem of the theory of probability. Uspekhi Mat. Nauk 3(3), 187–194 (1948)
Grama, I., Lauvergnat, R., Le Page, É.: Limit theorems for affine Markov walks conditioned to stay positive. Ann. Inst. Henri Poincaré Probabilités et Statistiques 54(1), 529–568 (2018)
Grama, I., Lauvergnat, R., Le Page, É.: Limit theorems for Markov walks conditioned to stay positive under a spectral gap assumption. Ann. Probab. 46(4), 1807–1877 (2018)
Grama, I., Le Page, É.: Bounds in the local limit theorem for a random walk conditioned to stay positive. In: Modern Problems of Stochastic Analysis and Statistics. Springer Proceedings in Mathematics & Statistics, pp. 103–130. Springer (2017)
Grama, I., Le Page, É., Peigné, M.: On the rate of convergence in the weak invariance principle for dependent random variables with application to Markov chains. Colloq. Math. 134(1), 1–55 (2014)
Grama, I., Le Page, É., Peigné, M.: Conditioned limit theorems for products of random matrices. Probab. Theory Relat. Fields 168(3–4), 601–639 (2017)
Guivarc’h, Y., Hardy, J.: Théorèmes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d’Anosov. Ann. IHP Probab. Stat. 24, 73–98 (1988)
Hennion, H., Hervé.: Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Mathematics, vol. 1766. Springer Berlin Heidelberg New York (2001)
Iglehart, D.L.: Functional central limit theorems for random walks conditioned to stay positive. Ann. Probab. 2(4), 608–619 (1974)
Iglehart, D.L.: Random walks with negative drift conditioned to stay positive. J. Appl. Probab. 11(4), 742–751 (1974)
Ionescu Tulcea, C.T., Marinescu, G.: Théorie ergodique pour des classes d’opérations non complètement continues. Ann. Math. 52(1), 140–147 (1950)
Kato, T.: Perturbation Theory for Linear Operators. Springer Berlin Heidelberg, Berlin (1976)
Kolmogorov, A.N.: A local limit theorem for classical Markov chains. Izv. Akad. Nauk SSSR, Ser. Math. 13(4), 281–300 (1949)
Le Page, E.: Théorèmes limites pour les produits de matrices aléatoires. In: Probability Measures on Groups, pp. 258–303. Springer, Berlin, Heidelberg (1982). https://doi.org/10.1007/BFb0093229
Le Page, É., Peigné, M.: A local limit theorem on the semi-direct product of \({\mathbb{R}}^{*+}\) and \({\mathbb{R}}^d\). Ann. Inst. Henri Poincare (B) Probab. Stat. 33, 223–252 (1997)
Nagaev, S.V.: Some limit theorems for stationary Markov chains. Theory Probab. Appl. 2(4), 378–406 (1954)
Nagaev, S.V.: More exact statement of limit theorems for homogeneous Markov chains. Theory Probab. Appl. 6(1), 62–81 (1961)
Presman, E.: Boundary problems for sums of lattice random variables, defined on a finite regular Markov chain. Theory Probab. Appl. 12(2), 323–328 (1967)
Presman, E.: Methods of factorization and a boundary problems for sums of random variables defined on a Markov chain. Izv. Akad. Nauk SSSR 33, 861–990 (1969)
Spitzer, F.: Principles of Random Walk. The University Series in Higher Mathematics. D. Van Nostrand, Princeton (1964)
Stone, C.: A local limit theorem for nonlattice multi-dimensional distribution functions. Ann. Math. Stat. 36(2), 546–551 (1965)
Varopoulos, NTh: Potential theory in conical domains. Math. Proc. Camb. Philos. Soc. 125(2), 335–384 (1999)
Varopoulos, NTh: Potential theory in conical domains. II. Math. Proc. Camb. Philos. Soc. 129(2), 301–320 (2000)
Vatutin, V.A., Wachtel, V.: Local probabilities for random walks conditioned to stay positive. Probab. Theory Relat. Fields 143(1–2), 177–217 (2008)
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Appendix
Appendix
1.1 The non degeneracy of the Markov walk
In [14], it is proved that the statements of Propositions 2.1–2.3 hold under more general assumptions (see Hypotheses M1-M5 of [14]). We will link these assumptions to our Hypotheses M1–M3. The assumptions M1-M3 in [14], with the Banach space \({\mathscr {C}}\), are well known consequences of Hypothesis M1 of this paper. Hypothesis M4 in [14] is also obvious with \(N=N_1 = \cdots = 0\). By Hypothesis M2, to obtain Hypothesis M5 of [14], it remains only to prove that \(\sigma \) defined by (2.2) is strictly positive. First we give a necessary and sufficient condition. Recall that the words path and orbit are defined in Sect. 4.
Lemma 10.1
Assume Hypothesis M1. The following statements are equivalent:
- 1.
The Cesáro mean of f on the orbits is constant: there exists \(m \in {\mathbb {R}}\) such that for any orbit \(x_0,\dots ,x_n\) we have
$$\begin{aligned} f(x_0) + \cdots + f(x_n) = (n+1)m. \end{aligned}$$ - 2.
There exist a constant \(m \in {\mathbb {R}}\) and a function \(h \in {\mathscr {C}}\) such that for any \((x,x') \in {\mathbb {X}}^2\),
$$\begin{aligned} {\mathbf {P}}(x,x') f(x') = {\mathbf {P}}(x,x') \left( h(x)-h(x')+m \right) . \end{aligned}$$ - 3.
The following real \({{\tilde{\sigma }}}^2\) is equal to 0
$$\begin{aligned} {{\tilde{\sigma }}}^2 = {\varvec{\nu }} \left( f^2 \right) - {\varvec{\nu }} \left( f \right) ^2 + 2 \sum _{n=1}^{+\infty } \left[ {\varvec{\nu }} \left( f {\mathbf {P}}^n f \right) - {\varvec{\nu }} \left( f \right) ^2 \right] = 0. \end{aligned}$$
Proof
The point 1 implies the point 2 Suppose that the point 1 holds. Fix \(x_0 \in {\mathbb {X}}\) and set \(h(x_0)= 0\). For any \(x \in {\mathbb {X}}\), we define h(x) in the following way: for any path \(x_0,x_1,\dots ,x_n,x\) in \({\mathbb {X}}\), we set
We shall verify that h is well defined. By Hypothesis M1, we can find at least a path to define h(x). Now we have to check that this definition does not depend on the choice of the path. Let \(x_0,x_1,\dots ,x_p,x\) and \(x_0,y_1,\dots ,y_q,x\) be two paths. By Hypothesis M1, there exists a path \(x,z_1, \dots , z_n,x_0\) in \({\mathbb {X}}\) between x and \(x_0\). Since \(x_0,x_1,\dots ,x_p,x,z_1,\dots ,z_n\) and \(x_0,y_1,\dots ,y_p,x,z_1,\dots ,z_n\) are two orbits, by the point 1, we have
and so the function h is well defined on \({\mathbb {X}}\). Now let \((x,x') \in {\mathbb {X}}^2\) such that \({\mathbf {P}}(x,x') > 0\). By Hypothesis M1, there exists \(x_0,x_1, \dots , x_n,x\) a path between \(x_0\) and x. Since
by the definition of h, we have
In particular
The point 2 implies the point 1 Suppose that the point 2 holds and let \(x_0,\dots ,x_n\) be an orbit. Using the point 2,
and the point 1 follows.
The point 2 implies the point 3 Suppose that the point 2 holds. Denote by \({{\tilde{f}}}\) the \({\varvec{\nu }}\)-centred function:
By the point 2, for any \(x\in {\mathbb {X}}\),
Using the fact that \({\varvec{\nu }}\) is \({\mathbf {P}}\)-invariant, we obtain that \({\varvec{\nu }} \left( {{\tilde{f}}} \right) = 0 = m-{\varvec{\nu }}(f)\) and so,
Consequently, by (10.2), \({\mathbf {P}}^n {{\tilde{f}}} = {\mathbf {P}}^{n-1} h - {\mathbf {P}}^n h\) for any \(n \geqslant 1\) and therefore,
Let
be the solution of the Poisson equation \({{\tilde{\varTheta }}} - {\mathbf {P}} {{\tilde{\varTheta }}} = {{\tilde{f}}}\), which by (2.1), is well defined. Taking the limit as \(n \rightarrow +\infty \) in (10.4) and using (2.1),
Therefore, for any \((x,x') \in {\mathbb {X}}^2\),
Using the point 2 and (10.3), it follows that
for any \((x,x') \in {\mathbb {X}}^2\) such that \({\mathbf {P}}(x,x') > 0\). Moreover,
Since \({\varvec{\nu }}\) is \({\mathbf {P}}\)-invariant,
By (10.5), we conclude that \({{\tilde{\sigma }}}^2=0\).
The point 3 implies the point 2 Suppose that the point 3 holds. By (10.6), for any \((x,x') \in {\mathbb {X}}\) such that \({\mathbf {P}}(x,x')>0\) we have
Let \(h = {\mathbf {P}} {{\tilde{\varTheta }}}\). Since \({{\tilde{\varTheta }}}\) is the solution of the Poisson equation,
By the definition of \({{\tilde{f}}}\) in (10.1), for any \((x,x') \in {\mathbb {X}}\) such that \({\mathbf {P}}(x,x')>0\),
with \(m = {\varvec{\nu }}(f)\). \(\square \)
Note that under Hypothesis M2, Lemma 10.1 can be rewritten as follows.
Lemma 10.2
Assume Hypotheses M1 and M2. The following statements are equivalent:
- 1.
The mean of f on the orbits is equal to zero: for any orbit \(x_0,\dots ,x_n\), we have
$$\begin{aligned} f(x_0) + \cdots + f(x_n) = 0. \end{aligned}$$ - 2.
There exists a function \(h \in {\mathscr {C}}\) such that for any \((x,x') \in {\mathbb {X}}^2\),
$$\begin{aligned} {\mathbf {P}}(x,x') f(x') = {\mathbf {P}}(x,x') \left( h(x)-h(x') \right) . \end{aligned}$$ - 3.
The real \(\sigma ^2\) is equal to 0:
$$\begin{aligned} \sigma ^2 = {\varvec{\nu }} \left( f^2 \right) + 2 \sum _{n=1}^{+\infty } {\varvec{\nu }} \left( f {\mathbf {P}}^n f \right) = 0. \end{aligned}$$
Now we prove that the Hypothesis M3 (the “non-lattice” condition), implies that the Markov walk has non-zero asymptotic variance.
Lemma 10.3
Under Hypotheses M1–M3, we have
Proof
We proceed by reductio ad absurdum. Suppose that \(\sigma ^2 = 0\). By Lemma 10.2, for any orbit \(x_0,\dots ,x_n\), we have
which implies the negation of Hypothesis M3 with \(\theta = a = 0\). \(\square \)
1.2 Strong approximation
Let \((B_t)_{t\geqslant 0}\) be the standard Brownian motion on \({\mathbb {R}}\) defined on the probability space \((\varOmega , {\mathscr {F}}, {\mathbb {P}})\). Consider the exit time
where \(\sigma \) is defined by (2.2). It is proved in Grama, Le Page and Peigné [16] that there is a version of the Markov walk \((S_n)_{n\geqslant 0}\) and of the standard Brownian motion \((B_t)_{t\geqslant 0}\) living on the same probability space which are close enough in the following sense:
Proposition 10.4
There exists \(\varepsilon _0 >0\) such that, for any \(\varepsilon \in (0,\varepsilon _0]\), \(x\in {\mathbb {X}}\) and \(n\geqslant 1\), without loss of generality (on an extension of the initial probability space) one can reconstruct the sequence \((S_n)_{n\geqslant 0}\) with a continuous time Brownian motion\((B_t)_{t\in {\mathbb {R}}_{+} }\), such that
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Grama, I., Lauvergnat, R. & Le Page, É. Conditioned local limit theorems for random walks defined on finite Markov chains. Probab. Theory Relat. Fields 176, 669–735 (2020). https://doi.org/10.1007/s00440-019-00948-8
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DOI: https://doi.org/10.1007/s00440-019-00948-8
Keywords
- Markov chain
- Exit time
- Conditioned local limit theorem
- Duality
Mathematics Subject Classification
- 60J10
- 60F05