Large deviations in discrete-time renewal theory

Abstract

We establish sharp large deviation principles for cumulative rewards associated with a discrete-time renewal model, supposing that each renewal involves a broad-sense reward taking values in a real separable Banach space. The framework we consider is the pinning model of polymers, which amounts to a Gibbs change of measure of a classical renewal process and includes it as a special case. We first tackle the problem in a constrained pinning model, where one of the renewals occurs at a given time, by an argument based on convexity and super-additivity. We then transfer the results to the original pinning model by resorting to conditioning.

Introduction

Renewal models are widespread tools of probability, finding application in queueing theory [1], insurance [15], and finance [40] among others. A renewal model describes some event that occurs at the renewal times $T_1,T_2,\dots$ involving the rewards $X_1,X_2,\dots$ respectively. If $S_1,S_2,\dots$ are the waiting times for a new occurrence of the event, then the renewal time $T_i$ can be expressed for each $i\ge 1$ in terms of waiting times as $T_i=S_1+\cdots +S_i$. This paper deals with cumulative rewards in renewal models with waiting times taking discrete values and rewards taking values in a Banach space. Specifically, we assume that the waiting time and reward pairs $(S_1,X_1),(S_2,X_2),\dots$ form an independent and identically distributed sequence of random variables on a probability space $(\Omega ,\mathcal{F},\mathbb{P})$, the waiting times being valued in $\{1,2,\dots \}\cup \left\{\infty \right\}$ and the rewards being valued in a real separable Banach space $(\mathcal{X},\Vert \cdot \Vert )$ equipped with the Borel $\sigma$-field $\mathcal{B}\left(\mathcal{X}\right)$. Any dependence between $X_i$ and $S_i$ is allowed. The cumulative reward by the integer time $t\ge 0$ is the random variable $W_t≔{\sum }_{i\ge 1}{X}_i{\mathbb{1}}_{\{T_i\le t\}}$, which is measurable because $\mathcal{X}$ is separable [31]. The stochastic process $t\mapsto W_t$ is the so-called renewal–reward process or compound renewal process, which plays an important role in applications [1], [15], [40]. The strong law of large numbers can be proved for a renewal–reward process under the optimal hypotheses $\mathbb{E}\left[S_1\right]<+\infty$ and $\mathbb{E}[\Vert X_1\Vert ]<+\infty$, $\mathbb{E}$ denoting expectation with respect to the law $\mathbb{P}$, by combining the argument of renewal theory [1] with the classical strong law of large numbers of Kolmogorov in separable Banach spaces [31].

In this paper we characterize the large fluctuations of the cumulative reward $W_t$ by establishing large deviation principles that generalize the Cramér’s theorem to discrete-time renewal models. Cramér’s theorem describes the large fluctuations of non-random sums of random variables, such as the total reward versus the number of renewals $n$ given by ${\sum }_{i=1}^n{X}_i$. It involves the rate function $I_{\mathrm{C}}$ that maps each point $w\in \mathcal{X}$ in the extended real number $I_{\mathrm{C}}\left(w\right)≔{sup}_{\phi \in {\mathcal{X}}^{\star }}\{\phi \left(w\right)-ln\mathbb{E}\left[e^{\phi \left(X_1\right)}\right]\}$, where ${\mathcal{X}}^{\star }$ is the topological dual of $\mathcal{X}$. The following sharp form of Cramér’s theorem has been obtained by Bahadur and Zabell [2] through an argument based on convexity and sub-additivity.

Cramér’s theorem

The following conclusions hold:

• the function $I_{\mathrm{C}}$ is lower semicontinuous and proper convex;

• if $G\subseteq \mathcal{X}$ is open, then

  $$\underset{n\uparrow \infty }{lim\; inf}\frac{1}{n}ln\mathbb{P}\left[\frac{1}{n}\sum _{i=1}^n{X}_i\in G\right]\ge -\underset{w\in G}{inf}\left\{I_{\mathrm{C}}\left(w\right)\right\};$$

• if $F\subseteq \mathcal{X}$ is compact, open convex, or closed convex, then

  $$\underset{n\uparrow \infty }{lim\; sup}\frac{1}{n}ln\mathbb{P}\left[\frac{1}{n}\sum _{i=1}^n{X}_i\in F\right]\le -\underset{w\in F}{inf}\left\{I_{\mathrm{C}}\left(w\right)\right\}.$$

  Furthermore, if $\mathcal{X}$ is finite-dimensional, then this bound is valid for any closed set $F$ provided that $\mathbb{E}\left[e^{\xi \Vert X_1\Vert }\right]<+\infty$ for some number $\xi >0$.

Earlier, Donsker and Varadhan [16] proved Cramér’s theorem under the stringent exponential moment condition $\mathbb{E}\left[e^{\xi \Vert X_1\Vert }\right]<+\infty$ for all $\xi >0$. Importantly, they showed that under this condition the upper bound in part (c) holds for any closed set $F$ even when $\mathcal{X}$ is infinite-dimensional.

Along with the use as stochastic processes, discrete-time renewal models find application in equilibrium statistical physics with a different interpretation of the time coordinate. In particular, they are employed in studying the phenomenon of polymer pinning, whereby a polymer consisting of $t\ge 1$ monomers is pinned by a substrate at the monomers $T_1,T_2,\dots$ that represent renewed events along the polymer chain [24], [27]. Supposing that the monomer $T_i$ contributes an energy $-v\left(S_i\right)$ provided that $T_i\le t$, $v$ being a real function over $\{1,2,\dots \}\cup \left\{\infty \right\}$ called the potential, the state of the polymer is described by the perturbed law ${\mathbb{P}}_t$ defined on the measurable space $(\Omega ,\mathcal{F})$ by the Gibbs change of measure

$$\frac{d{\mathbb{P}}_t}{d\mathbb{P}}≔\frac{e^{H_t}}{Z_t},$$

where $H_t≔{\sum }_{i\ge 1}v\left(S_i\right){\mathbb{1}}_{\{T_i\le t\}}$ is the Hamiltonian and the normalizing constant $Z_t≔\mathbb{E}\left[e^{H_t}\right]$ is the partition function. The model $(\Omega ,\mathcal{F},{\mathbb{P}}_t)$ is called the pinning model (PM) and generalizes the original renewal model corresponding to the potential $v=0$. The theory of large deviations we develop in this paper is framed within the PM supplied with the hypotheses of aperiodicity and extensivity. The waiting time distribution $p≔\mathbb{P}[S_1=\cdot ]$ is said to be aperiodic if $\mathbb{P}[S_1<\infty ]>0$ and there is no proper sublattice of $\{1,2,\dots \}$ containing the support of $p$. We point out that a generic $p$ can be made aperiodic when $\mathbb{P}[S_1<\infty ]>0$ by simply changing the time unit.

Assumption 1.1

The waiting time distribution $p$ is aperiodic.

We say that the potential $v$ is extensive if there exists a real number $z_o$ such that $e^{v\left(s\right)}p\left(s\right)\le e^{z_{o}s}$ for all $s$. For instance, any potential $v$ with the property that ${sup}_{s\ge 1}\{v\left(s\right)∕s\}<+\infty$ is extensive. Extensive potentials are the only that serve equilibrium statistical physics, where the partition function $Z_t\ge \mathbb{E}\left[e^{H_t}{\mathbb{1}}_{\{S_1=t\}}\right]=e^{v\left(t\right)}p\left(t\right)$ is expected to grow exponentially in $t$ in order to define the free energy [24], [27].

Assumption 1.2

The potential $v$ is extensive.

Together with the PM we consider the constrained pinning model (CPM) where the last monomer is always pinned by the substrate [24], [27]. It corresponds to the law ${\mathbb{P}}_t^c$ defined on the measurable space $(\Omega ,\mathcal{F})$ through the change of measure

$$\frac{d{\mathbb{P}}_t^c}{d\mathbb{P}}≔\frac{U_t{e}^{H_t}}{Z_t^c},$$

where $U_t≔{\sum }_{i\ge 1}{\mathbb{1}}_{\{T_i=t\}}$ is the renewal indicator, which takes value 1 if $t$ is a renewal and value 0 otherwise, and $Z_t^c≔\mathbb{E}\left[U_t{e}^{H_t}\right]$ is the partition function. Our interest in the CPM is twofold. On the one hand, it turns out to be an effective mathematical tool to tackle the PM. Indeed, we can obtain a large deviation principle within the CPM by an argument based on convexity and super-additivity, and then transfer it to the PM by conditioning. The mentioned argument is a generalization of the approach to Cramér’s theorem by Bahadur and Zabell [2], which in turn can be traced back to the method of Ruelle [42] and Lanford [30] for proving the existence of various thermodynamic limits. On the other hand, the CPM is a significant framework in itself because it is the mathematical skeleton of the Poland–Scheraga model of DNA denaturation and of some relevant lattice models of statistical mechanics, as discussed by the author in Ref. [46] where use of the theory developed in the present paper is made. These models are the cluster model of fluids proposed by Fisher and Felderhof, the model of protein folding introduced independently by Wako and Saitô first and Muñoz and Eaton later, and the model of strained epitaxy considered by Tokar and Dreyssé. The macroscopic observables that enter the thermodynamic description of these systems turn out to be cumulative rewards corresponding to rewards of the order of magnitude of the waiting times [46].

Before introducing our main results, we must say that the CPM is not well-defined a priori. In fact, it may happen with full probability that the time $t$ is not a renewal, so that $Z_t^c=0$. However, Assumption 1.1 resulting in $Z_t^c>0$ for every sufficiently large $t$ settles the problem at least for all those $t$. To verify this fact, we observe that aperiodicity of $p$ entails that there exist $m$ coprime integers ${\sigma }_1,\dots ,{\sigma }_m$ such that $p\left({\sigma }_l\right)>0$ for each $l$. The bound $Z_t^c\ge \mathbb{E}\left[U_t{e}^{H_t}{\prod }_{i=1}^n{\mathbb{1}}_{\{S_i=s_i\}}\right]={\prod }_{i=1}^n{e}^{v\left(s_i\right)}p\left(s_i\right)$ if $t={\sum }_{i=1}^n{s}_i$ yields $Z_t^c>0$ whenever $t$ is an integer conical combination of ${\sigma }_1,\dots ,{\sigma }_m$. On the other hand, the Frobenius number $t_c\ge 0$ associated with ${\sigma }_1,\dots ,{\sigma }_m$ is finite since these integers are coprime and by definition any $t>t_c$ can be expressed as an integer conical combination of them. It follows that $Z_t^c>0$ for all $t>t_c$.

This section reports the main results of the paper. In the sequel, Assumption 1.1, Assumption 1.2 are tacitly supposed to be satisfied and the topological dual ${\mathcal{X}}^{\star }$ of $\mathcal{X}$ is understood as a Banach space with the norm induced by $\Vert \cdot \Vert$. Let $z$ be the function that maps each linear functional $\phi \in {\mathcal{X}}^{\star }$ in the extended real number $z\left(\phi \right)$ defined by

$$z\left(\phi \right)≔inf\left\{\zeta \in \mathbb{R}:\mathbb{E}\left[e^{\phi \left(X_1\right)+v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]\le 1\right\},$$

where the infimum over the empty set is customarily interpreted as $+\infty$. The following proposition puts this function into context by relating $z$ to the scaled cumulant generating function of $W_t$ within the CPM. According to this proposition, $z\left(0\right)$ turns out to be the free energy of the CPM [24], [27] and, more in general, $z\left(\phi \right)$ can be regarded as the free energy of a CPM with the (possibly non-extensive) potential $v+ln\mathbb{E}\left[e^{\phi \left(X_1\right)}\right|S_1=\cdot ]$.

Proposition 1.1

The function $z$ is proper convex and lower semicontinuous. The following limit holds for every $\phi \in {\mathcal{X}}^{\star }$:

$$\underset{t\uparrow \infty }{lim}\frac{1}{t}ln\mathbb{E}\left[U_t{e}^{\phi \left(W_t\right)+H_t}\right]=z\left(\phi \right).$$

Denoting the expectation with respect to the law ${\mathbb{P}}_t^c$ by ${\mathbb{E}}_t^c$, Proposition 1.1 entails that ${lim}_{t\uparrow \infty }(1∕t)ln{\mathbb{E}}_t^c\left[e^{\phi \left(W_t\right)}\right]=z\left(\phi \right)-z\left(0\right)$ for all $\phi \in {\mathcal{X}}^{\star }$, so that $z-z\left(0\right)$ is exactly the scaled cumulant generating function of $W_t$ within the CPM. We stress that the number $z\left(0\right)$ is finite. Indeed, we have $\mathbb{E}\left[e^{v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]>1$ for all sufficiently negative $\zeta$ as $\mathbb{P}[S_1<\infty ]>0$ by Assumption 1.1 and, at the same time, $\mathbb{E}\left[e^{v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]={\sum }_{s\ge 1}{e}^{v\left(s\right)-\zeta s}p\left(s\right)\le 1$ for all $\zeta \ge z_o+ln2$, $z_o$ being the number introduced by Assumption 1.2. The function $z$ is finite everywhere in the following case, which is relevant for statistical mechanics as it comprises the macroscopic observables that enter the thermodynamic description of the system [46].

Example 1.1

The function $z$ is finite everywhere if the reward $X_1$ is dominated by the waiting time $S_1$ in the sense that $\Vert X_1\Vert \le M{S}_1$ with full probability for some constant $M<+\infty$. This follows from the facts that, for any given $\phi \in {\mathcal{X}}^{\star }$, $\mathbb{E}\left[e^{\phi \left(X_1\right)+v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]\ge \mathbb{E}\left[e^{-M\Vert \phi \Vert S_1+v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]>1$ for all sufficiently negative $\zeta$, as $\mathbb{P}[S_1<\infty ]>0$ by Assumption 1.1, and $\mathbb{E}\left[e^{\phi \left(X_1\right)+v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]\le {\sum }_{s\ge 1}{e}^{M\Vert \phi \Vert s+v\left(s\right)-\zeta s}p\left(s\right)\le 1$ for all $\zeta \ge z_o+M\Vert \phi \Vert +ln2$ with $z_o$ given by Assumption 1.2.

We use the function $z$ to construct a rate function. Let $I$ be the Fenchel–Legendre transform of $z-z\left(0\right)$, which associates every point $w\in \mathcal{X}$ with the extended real number $I\left(w\right)$ given by

$$I\left(w\right)≔\underset{\phi \in {\mathcal{X}}^{\star }}{sup}\left\{\phi \left(w\right)-z\left(\phi \right)+z\left(0\right)\right\}.$$

The following theorem extends Cramér’s theorem to the cumulative reward $W_t$ with respect to the CPM and constitutes our first main result. It is proved together with Proposition 1.1 in Section 2.

Theorem 1.1

The following conclusions hold:

• the function $I$ is lower semicontinuous and proper convex;

• if $G\subseteq \mathcal{X}$ is open, then

  $$\underset{t\uparrow \infty }{lim\; inf}\frac{1}{t}ln{\mathbb{P}}_t^c\left[\frac{W_t}{t}\in G\right]\ge -\underset{w\in G}{inf}\left\{I\left(w\right)\right\};$$

• if $F\subseteq \mathcal{X}$ is compact, open convex, closed convex, or any convex set in $\mathcal{B}\left(\mathcal{X}\right)$ when $\mathcal{X}$ is finite-dimensional, then

  $$\underset{t\uparrow \infty }{lim\; sup}\frac{1}{t}ln{\mathbb{P}}_t^c\left[\frac{W_t}{t}\in F\right]\le -\underset{w\in F}{inf}\left\{I\left(w\right)\right\}.$$

  Furthermore, if $\mathcal{X}$ is finite-dimensional, then this bound is valid for any closed set $F$ provided that $\mathbb{E}\left[e^{\xi \Vert X_1\Vert +v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]<+\infty$ for some numbers $\zeta \ge 0$ and $\xi >0$.

The lower bound in part (b) and the upper bound in part (c) are called, respectively, large deviation lower bound and large deviation upper bound [14], [26]. When a lower semicontinuous function $I$ exists so that the large deviation lower bound holds for each open set $G$ and the large deviation upper bound holds for each compact set $F$, then $W_t$ is said to satisfy a weak large deviation principle (weak LDP) with rate function $I$ [14], [26]. If the large deviation upper bound holds more generally for every closed set $F$, then $W_t$ is said to satisfy a full large deviation principle (full LDP) [14], [26]. Theorem 1.1 states that the cumulative reward $W_t$ satisfies a weak LDP with rate function $I$ given by (2) within the CPM. If in addition $\mathcal{X}$ is finite-dimensional and the exponential moment condition $\mathbb{E}\left[e^{\xi \Vert X_1\Vert +v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]<+\infty$ is fulfilled for some $\zeta \ge 0$ and $\xi >0$, as certainly occurs in Example 1.1 for any $\xi >0$ and $\zeta >M\xi +z_o$, then $W_t$ satisfies a full LDP. Regarding the validity of a full LDP for general infinite-dimensional Banach spaces $\mathcal{X}$, finding sufficient conditions is a harder problem that will be the focus of future studies. Trying to sketch an analogy with Cramér’s theorem and the work by Donsker and Varadhan [16], one should probably investigate situations where there exists $\zeta \ge 0$ such that $\mathbb{E}\left[e^{\xi \Vert X_1\Vert +v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]<+\infty$ for all $\xi >0$.

Let us move now to the PM, where there is no constraint on the last monomer. At variance with the CPM, the scaled cumulant generating function of $W_t$ may not exist in the PM, but the following proposition, which is proved in Section 3, shows that at least some bounds hold true. Set ${\ell }_{\mathrm{i}}≔{lim\; inf}_{t\uparrow \infty }(1∕t)ln\mathbb{P}[S_1>t]$ and ${\ell }_{\mathrm{s}}≔{lim\; sup}_{t\uparrow \infty }(1∕t)ln\mathbb{P}[S_1>t]$, and bear in mind that $-\infty \le {\ell }_{\mathrm{i}}\le {\ell }_{\mathrm{s}}\le 0$.

Proposition 1.2

The following bounds hold for all $\phi \in {\mathcal{X}}^{\star }$:

$$z\left(\phi \right)\vee {\ell }_{\mathrm{i}}\le \underset{t\uparrow \infty }{lim\; inf}\frac{1}{t}ln\mathbb{E}\left[e^{\phi \left(W_t\right)+H_t}\right]\le \underset{t\uparrow \infty }{lim\; sup}\frac{1}{t}ln\mathbb{E}\left[e^{\phi \left(W_t\right)+H_t}\right]\le z\left(\phi \right)\vee {\ell }_{\mathrm{s}}.$$

Denoting by ${\mathbb{E}}_t$ the expectation with respect to ${\mathbb{P}}_t$, Proposition 1.2 entails that the limit ${lim}_{t\uparrow \infty }(1∕t)ln{\mathbb{E}}_t\left[e^{\phi \left(W_t\right)}\right]$ exists, and equals $z\left(\phi \right)\vee {\ell }_{\mathrm{s}}-z\left(0\right)\vee {\ell }_{\mathrm{s}}$, if either ${\ell }_{\mathrm{i}}={\ell }_{\mathrm{s}}$ or $z\left(\phi \right)\ge {\ell }_{\mathrm{s}}$. Thus, the scaled cumulant generating function of $W_t$ with respect to the PM is defined if either ${\ell }_{\mathrm{i}}={\ell }_{\mathrm{s}}$, which includes the case ${\ell }_{\mathrm{s}}=-\infty$, or the condition $z\left(\phi \right)\ge {\ell }_{\mathrm{s}}>-\infty$ is met for all $\phi \in {\mathcal{X}}^{\star }$, as in the following example.

Example 1.2

The bound $z\left(\phi \right)\ge {\ell }_{\mathrm{s}}>-\infty$ holds for all $\phi \in {\mathcal{X}}^{\star }$ if $\mathbb{P}[S_1<\infty ]=1$, ${lim\; inf}_{s\uparrow \infty }v\left(s\right)∕s=0$, and there exists a positive real function $g$ on $\{1,2,\dots \}\cup \left\{\infty \right\}$ such that ${lim}_{s\uparrow \infty }g\left(s\right)∕s=0$ and $\Vert X_1\Vert \le g\left(S_1\right)$ with full probability. Indeed, given any $\zeta <{\ell }_{\mathrm{s}}$, under these hypotheses one can find $ϵ>0$ such that $\zeta +ϵ<{\ell }_{\mathrm{s}}\le 0$ and $-\Vert \phi \Vert g\left(s\right)+v\left(s\right)\ge -ϵs$ for all sufficiently large $s$. Then, $\mathbb{E}\left[e^{\phi \left(X_1\right)+v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]\ge {\sum }_{s\ge 1}{e}^{-\Vert \phi \Vert g\left(s\right)+v\left(s\right)-\zeta s}p\left(s\right)\ge {\sum }_{s>t}{e}^{-(\zeta +ϵ)s}p\left(s\right)\ge e^{-(\zeta +ϵ)t}\mathbb{P}[S_1>t]$ for all sufficiently large $t$ as $\mathbb{P}[S_1=\infty ]=0$. It follows that $\mathbb{E}\left[e^{\phi \left(X_1\right)+v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]=+\infty$ since $\zeta +ϵ<{\ell }_{\mathrm{s}}$, which results in $z\left(\phi \right)\ge {\ell }_{\mathrm{s}}$ according to definition (1).

In order to establish large deviation bounds with respect to the PM, it is convenient to distinguish the case ${\ell }_{\mathrm{s}}=-\infty$ from the case ${\ell }_{\mathrm{s}}>-\infty$. The following theorem, which represents our second main result, provides weak and full LDPs for the renewal–reward process $t\mapsto W_t$ with respect to the PM when ${\ell }_{\mathrm{s}}=-\infty$. The proof is given in Section 3.

Theorem 1.2

Assume ${\ell }_{\mathrm{s}}=-\infty$. The following conclusions hold:

• if $G\subseteq \mathcal{X}$ is open, then

  $$\underset{t\uparrow \infty }{lim\; inf}\frac{1}{t}ln{\mathbb{P}}_t\left[\frac{W_t}{t}\in G\right]\ge -\underset{w\in G}{inf}\left\{I\left(w\right)\right\};$$

• if $F\subseteq \mathcal{X}$ is compact, then

  $$\underset{t\uparrow \infty }{lim\; sup}\frac{1}{t}ln{\mathbb{P}}_t\left[\frac{W_t}{t}\in F\right]\le -\underset{w\in F}{inf}\left\{I\left(w\right)\right\}.$$

  If $F\subseteq \mathcal{X}$ is open convex, closed convex, or any convex set in $\mathcal{B}\left(\mathcal{X}\right)$ when $\mathcal{X}$ is finite-dimensional, then this bound is valid whenever $I\left(0\right)<+\infty$. Furthermore, if $\mathcal{X}$ is finite-dimensional, then it is valid for any closed set $F$ provided that$\mathbb{E}\left[e^{\xi \Vert X_1\Vert +v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]<+\infty$ for some numbers $\zeta \ge 0$ and $\xi >0$.

In general, the large deviation upper bound in part (b) cannot be extended to convex sets if ${\ell }_{\mathrm{s}}=-\infty$ and $I\left(0\right)=+\infty$. Examples with an open convex set and a closed convex set where such bound fails will be shown at the end of Section 3.

The case ${\ell }_{\mathrm{s}}>-\infty$ is more involved and calls for two rate functions, $I_{\mathrm{i}}$ and $I_{\mathrm{s}}$, which are defined for each $w\in \mathcal{X}$ by the formulas

$$I_{\mathrm{i}}\left(w\right)≔\underset{\phi \in {\mathcal{X}}^{\star }}{sup}\left\{\phi \left(w\right)-z\left(\phi \right)\vee {\ell }_{\mathrm{i}}+z\left(0\right)\vee {\ell }_{\mathrm{s}}\right\}$$

and

$$I_{\mathrm{s}}\left(w\right)≔\underset{\phi \in {\mathcal{X}}^{\star }}{sup}\left\{\phi \left(w\right)-z\left(\phi \right)\vee {\ell }_{\mathrm{s}}+z\left(0\right)\vee {\ell }_{\mathrm{i}}\right\}.$$

The following theorem, which is our third and last main result, provides large deviation bounds with respect to the PM when ${\ell }_{\mathrm{s}}>-\infty$. The proof is reported in Section 3.

Theorem 1.3

Assume ${\ell }_{\mathrm{s}}>-\infty$. The following conclusions hold:

• the functions $I_{\mathrm{i}}$ and $I_{\mathrm{s}}$ are lower semicontinuous and proper convex;

• if $G\subseteq \mathcal{X}$ is open, then

  $$\underset{t\uparrow \infty }{lim\; inf}\frac{1}{t}ln{\mathbb{P}}_t\left[\frac{W_t}{t}\in G\right]\ge -\underset{w\in G}{inf}\left\{I_{\mathrm{i}}\left(w\right)\right\};$$

• if $F\subseteq \mathcal{X}$ is compact, open convex, closed convex, or any convex set in $\mathcal{B}\left(\mathcal{X}\right)$ when $\mathcal{X}$ is finite-dimensional, then

  $$\underset{t\uparrow \infty }{lim\; sup}\frac{1}{t}ln{\mathbb{P}}_t\left[\frac{W_t}{t}\in F\right]\le -\underset{w\in F}{inf}\left\{I_{\mathrm{s}}\left(w\right)\right\}.$$

  Furthermore, if $\mathcal{X}$ is finite-dimensional, then this bound is valid for any closed set $F$ provided that $\mathbb{E}\left[e^{\xi \Vert X_1\Vert +v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]<+\infty$ for some numbers $\zeta \ge 0$ and $\xi >0$.

Theorem 1.3 states that the renewal–reward process $t\mapsto W_t$ satisfies a weak LDP with rate function $I_{\mathrm{s}}$ within the PM provided that $I_{\mathrm{i}}=I_{\mathrm{s}}$. The exponential moment condition $\mathbb{E}\left[e^{\xi \Vert X_1\Vert +v\left(S_1\right)-\zeta S_1}{\mathbb{1}}_{\{S_1<\infty \}}\right]<+\infty$ for some $\zeta \ge 0$ and $\xi >0$ gives a full LDP with rate function $I_{\mathrm{s}}$ when $\mathcal{X}$ is finite-dimensional and $I_{\mathrm{i}}=I_{\mathrm{s}}$. We have $I_{\mathrm{i}}=I_{\mathrm{s}}$ if ${\ell }_{\mathrm{i}}={\ell }_{\mathrm{s}}$, as expected in most applications, or if the condition $z\left(\phi \right)\ge {\ell }_{\mathrm{s}}$ is fulfilled for all $\phi \in {\mathcal{X}}^{\star }$, as in Example 1.2. In the latter case, $I_{\mathrm{i}}=I_{\mathrm{s}}=I$.

Large deviations for renewal–reward processes have been investigated by many authors over the past decades. Their attention has been focused on both discrete-time and continuous-time frameworks and, in most cases, on rewards taking real values. In order to fix ideas, when talking about renewal systems in the domain of time we think of a PM with waiting times satisfying $\mathbb{P}[S_1<\infty ]=1$ and potential $v=0$. An almost omnipresent hypothesis in previous works is the Cramér condition $\mathbb{E}\left[e^{\xi \Vert X_1\Vert +\xi S_1}\right]<+\infty$ for some number $\xi >0$.

The simplest example of renewal–reward process has unit rewards and corresponds to the counting renewal process $t\mapsto N_t≔{\sum }_{i\ge 1}{\mathbb{1}}_{\{T_i\le t\}}$. Glynn and Whitt [25] investigated the connection between LDPs of the inverse processes $t\mapsto N_t$ and $i\mapsto T_i$, providing a full LDP for $N_t$ under the Cramér condition. This condition was later relaxed by Duffield and Whitt [17]. Jiang [28] studied the large deviations of the extended counting renewal process $t\mapsto {\sum }_{i\ge 1}{\mathbb{1}}_{\{T_i\le i^{\alpha }t\}}$ with $\alpha \in [0,1)$ under the Cramér condition. Glynn and Whitt [25] and Duffield and Whitt [17], together with Puhalskii and Whitt [39], also investigated the connection between sample-path LDPs of the processes $t\mapsto N_t$ and $i\mapsto T_i$ under the Cramér condition.

Starting from sample-path LDPs of inverse and compound processes, Duffy and Rodgers-Lee [18] sketched a full LDP for renewal–reward processes with real rewards by means of the contraction principle under the stringent exponential moment condition $\mathbb{E}\left[e^{\xi \Vert X_1\Vert +\xi S_1}\right]<+\infty$ for all $\xi >0$. Some full LDPs for real renewal–reward processes were later proposed by Macci [35], [36] under existence and essentially smoothness of the scaled cumulant generating function, which allow for an application of the Gärtner–Ellis theorem [14], [26]. Essentially smoothness of the scaled cumulant generating function has been recently relaxed by Borovkov and Mogulskii [5], [6], which used the Cramér’s theorem to establish a full LDP under the Cramér condition. Under the Cramér condition, they [4], [7], [8] have also obtained sample-path LDPs for real renewal–reward processes.

A different approach based on empirical measures has been considered by Lefevere, Mariani, and Zambotti [32], which have investigated large deviations for the empirical measures of forward and backward recurrence times associated with a renewal process, and have then derived by contraction a full LDP for renewal–reward processes with rewards determined by the waiting times: $X_i≔f\left(S_i\right)$ for each $i$ with a bounded real function $f$. Later, Mariani and Zambotti [37] have developed a renewal version of Sanov’s theorem by studying the empirical law of rewards that take values in a generic Polish space. By appealing to the contraction principle, this result could give a full LDP for a renewal–reward process with rewards valued in a real separable Banach space, but only provided that the strong exponential moment condition $\mathbb{E}\left[e^{\xi \Vert X_1\Vert }\right]<+\infty$ for all $\xi >0$ is satisfied as discussed by Schied [43].

We conclude this brief review of previous contributions by mentioning that a moderate deviation principle for real renewal–reward processes was obtained by Tsirelson [45] under an exponential moment condition. Exact asymptotics for the counting renewal process and real renewal–reward processes has been investigated under the Cramér condition and several additional smoothness hypotheses by Serfozo [44], Kuczek and Crank [29], Chi [12], and Borovkov and Mogulskii [9], [10].

Previous works leave open the question of whether some large deviation principles free from exponential moment conditions can be established for renewal–reward processes, in the wake of the sharp version of Cramér’s theorem demonstrated by Bahadur and Zabell [2]. The present paper gives a positive answer to this question at the price of restricting to the discrete-time framework. Indeed, through Theorems 1.1, 1.2, and 1.3 we supply weak LDPs and large deviation upper bounds for measurable convex sets that are completely free from hypotheses. Moreover, when finite-dimensional rewards are considered, and when $\mathbb{P}[S_1<\infty ]=1$ and $v=0$ to make a comparison with previous studies, we provide full LDPs under the exponential moment condition $\mathbb{E}\left[e^{\xi \Vert X_1\Vert -\zeta S_1}\right]<+\infty$ for some numbers $\zeta \ge 0$ and $\xi >0$, which is weaker than the Cramér condition $\mathbb{E}\left[e^{\xi \Vert X_1\Vert +\xi S_1}\right]<+\infty$ for some $\xi >0$. For instance, rewards of Example 1.1 that define the macroscopic observables of statistical mechanics [46] always satisfy our weak exponential moment condition, whereas in general they do not fulfill the Cramér condition.

In order to drop exponential moment conditions, a novel approach with respect to past methods had to be devised to tackle the problem, and a new approach was suggested to us by the theory of polymer pinning [24], [27]. This new approach is based on super-additivity, but requires discrete time to be implemented. It came from here the need to focus on the discrete-time framework. In such framework, conditioning on the event that the last time is a renewal time is a meaningful procedure and enables a super-additivity property of renewal–reward processes to emerge. This procedure introduces a constrained model similarly to what is done with polymers. This way, we were able to find a successful strategy for investigating large deviations and we were naturally led to link renewal–reward processes to the PM and the CPM. Importantly, the CPM is not a merely mathematical tool to tackle the PM, but it also represents the renewal models of statistical mechanics [46], such as the Poland–Scheraga model, the Fisher–Felderhof model, the Wako–Saitô–Muñoz–Eaton model, and the Tokar–Dreyssé model. In this respect, the large deviation theory developed in this paper must be added to those already existing for other models of statistical mechanics, including the Curie–Weiss model [19], the Curie–Weiss–Potts model [13], the mean-field Blume–Emery–Griffiths model [21], and, to some extent, the Ising model as well as general Gibbs measures relative to an interaction potential [20], [22], [23], [38].

Going back for a moment to the domain of time with $\mathbb{P}[S_1<\infty ]=1$ and $v=0$, it is interesting to point out that Duffy and Rodgers-Lee [18], Lefevere, Mariani, and Zambotti [32], and Borovkov and Mogulskii [5], [6] found, with increasing level of generality, an apparently different rate function. They constructed the rate function for renewal–reward processes from the Cramér rate function ${\Upsilon }_{\mathrm{C}}$ of the waiting time and reward pair $(S_1,X_1)$, defined for each pair $(\beta ,w)\in \mathbb{R}\times \mathcal{X}$ by

$${\Upsilon }_{\mathrm{C}}(\beta ,w)≔\underset{(\zeta ,\phi )\in \mathbb{R}\times {\mathcal{X}}^{\star }}{sup}\left\{\phi \left(w\right)-\beta \zeta -ln\mathbb{E}\left[e^{\phi \left(X_1\right)-\zeta S_1}\right]\right\}.$$

Starting from ${\Upsilon }_{\mathrm{C}}$, they considered the function ${inf}_{\gamma >0}\left\{\gamma {\Upsilon }_{\mathrm{C}}(\cdot ∕\gamma ,\cdot ∕\gamma )\right\}$, whose lower-semicontinuous regularization $\Upsilon$ is given for every $(\beta ,w)\in \mathbb{R}\times \mathcal{X}$ by

Here $B_{w,\delta }≔\{u\in \mathcal{X}:\Vert u-w\Vert <\delta \}$ is the open ball of center $w$ and radius $\delta$. Duffy and Rodgers-Lee [18] dealt with the case $\mathcal{X}=\mathbb{R}$ and ${\ell }_{\mathrm{s}}=-\infty$ under a strong exponential moment condition, obtaining the rate function $\Lambda ≔\Upsilon (1,\cdot )$. In this case we have the rate function $I$ by Theorem 1.2 with $z\left(\phi \right)=inf\{\zeta \in \mathbb{R}:\mathbb{E}\left[e^{\phi \left(X_1\right)-\zeta S_1}\right]\le 1\}$ for all $\phi$ as $S_1<\infty$ with full probability and $v=0$. Lefevere, Mariani, and Zambotti [32] too found the rate function $\Lambda$ when $X_1=f\left(S_1\right)$ with a bounded real function $f$. This instance falls under the umbrella of Example 1.2, and so we get again the rate function $I$ by Theorem 1.3. Borovkov and Mogulskii [5], [6] studied the case $\mathcal{X}=\mathbb{R}$ under the Cramér condition. They obtained the rate function $\Lambda$ when ${\ell }_{\mathrm{s}}=-\infty$ and the rate function ${\Lambda }_{\mathrm{s}}≔{inf}_{\beta \in [0,1]}\{\Upsilon (\beta ,\cdot )-(1-\beta ){\ell }_{\mathrm{s}}\}$ when ${\ell }_{\mathrm{i}}={\ell }_{\mathrm{s}}>-\infty$ or $z\left(\phi \right)\ge {\ell }_{\mathrm{s}}>-\infty$ for all $\phi \in {\mathcal{X}}^{\star }$. In these cases we have the rate functions $I$ and $I_{\mathrm{s}}$, respectively, by Theorem 1.2, Theorem 1.3. Despite different expressions, our results are consistent with all the findings of these authors. Indeed, while uniqueness of the rate function [14], [26] suggests that there is at least some situation where $I=\Lambda$ and $I_{\mathrm{s}}={\Lambda }_{\mathrm{s}}$, a direct comparison shows that these identities hold in general, as established by the following lemma which is proved in the Appendix.

Lemma 1.1

Assume that $\mathbb{P}[S_1<\infty ]=1$ and that $v=0$. Then, the following conclusions hold for every $w\in \mathcal{X}$:

• $I\left(w\right)=\Lambda \left(w\right)≔\Upsilon (1,w)$;

• $I_{\mathrm{s}}\left(w\right)={\Lambda }_{\mathrm{s}}\left(w\right)≔{inf}_{\beta \in [0,1]}\{\Upsilon (\beta ,w)-(1-\beta ){\ell }_{\mathrm{s}}\}$ provided that ${\ell }_{\mathrm{s}}>-\infty$.

As a final remark, we stress that Borovkov and Mogulskii [5], [6] opted for not introducing two different rate functions for the large deviation lower and upper bounds, thus considering only problems where $I_{\mathrm{i}}=I_{\mathrm{s}}$. At variance with them, we decided to provide optimal large deviation bounds with possibly different rate functions in order to even address situations where the tail of the waiting time distribution is very oscillating. For instance, a physical renewal model giving rise to two possibly different rate functions has been found by Lefevere, Mariani, and Zambotti [33], [34] in the description of a free particle interacting with a heat bath.