On Asymptotics of Optimal Stopping Times

We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density. The objective of such problems is to find a procedure which maximizes the expected reward; this is often known as the"full information"problem. In this analysis, we obtain asymptotic expressions for the expectation and variance of the optimal stopping time as the number of drawn variables becomes large. In the case of distributions with infinite upper bound, the asymptotic behaviour of these statistics depends solely on the algebraic power of the probability distribution decay rate in the upper limit. In the case of densities with finite upper bound, the asymptotic behaviour these statistics depend on the algebraic form of the distribution near the finite upper bound. Explicit calculations are provided for several common probability density functions.

The optimal stopping rule τ N is While there exists extensive literature on the optimal stopping problem (see, for example, [2,3]), less attention has been paid to asymptotic properties of stopping times.In [7], the authors found the asymptotics of the expected value and the variance of the stopping time for a sequence of iid random variables having the standard uniform distribution; see also [4].In this paper, we are interested in finding asymptotics of E(τ N ) and Var(τ N ) as N → ∞ for general classes of pdfs.

Probability density functions on domains with infinite upper bound
Assume that y n is drawn from a continuous pdf f (y), which has unbounded support in the positive direction.This function has a cumulative distribution function (cdf) F (y).As we will be particularly interested in the asymptotic behaviour of the tail of the cdf, we define h(y) = 1 − F (y) for notational convenience.We restrict our attention to distributions with h(y) = o(y −1 ) in the large-y limit, which corresponds to pdfs having upper tails that satisfy f (y) = o(y −2 ) as y → ∞.
We note that if h(y) decays exponentially as y → ∞, this implies that f (y) must also decay exponentially in this limit.A similar observation is true for algebraically-decaying h(y) as y → ∞, which must arise from f (y) with algebraically decaying behaviour in this limit, or heavy-tailed distributions.These two cases must be studied separately.
We first obtain an expression for v n .The recurrence relation (1) implies We express this in terms of h(x), giving Recalling that h(y) = o(y −1 ) as y → ∞, we see that the integral expression converges and lim K→∞ Kh(K) = 0. Hence, the recurrence relation becomes

Large-n asymptotics of v n
The recurrence relation obtained in (2) can be used to directly compute the asymptotic behaviour of v n for many common pdfs, including all densities listed in Table 1.A general form of the asymptotic behaviour of v n in the limit that n → ∞ for broad classes of pdf may be derived using a WKB (Wentzel-Kramer-Brillouin), or Green-Liouville ansatz (see [9], for example).

Asymptotics of v n with Exponentially Decaying Tails
We are interested in the asymptotic behaviour of the recurrence relation as n → ∞.We therefore rescale n = s/δ, where 0 < δ ≪ 1 and s = O(1) as δ → 0. We abuse notation slightly to represent v n in terms of the rescaled variable, so that v n = v(s).This gives h(y)dy.
Expanding this using a Taylor series about δ = 0 gives in the limit that δ → 0. For convenience, we define the antiderivative of h(y) as H(y), such that H ′ (y) = h(y).This gives where we utilise standard notation for asymptotic equivalence.Recall that h(y) = o(y −1 ) as y → ∞, and hence its antiderivative H(y) also tends to zero in this limit.This expression is therefore a separable differential equation, with solution Here, we make the assumption that v becomes large in the limit δ → 0 (corresponding to n → ∞) and study the large-v asymptotics of the integral expression.We first pose a WKB ansatz which assumes the integral can be expressed in terms of exponentials and power series.This ansatz takes the form where p(v) and q(v) are polynomial expressions in v, and q(v) → ∞ as v → ∞.We assume that p(v) ∼ av b and q(v) ∼ cv d , with a = 0 and c, d > 0, as v → ∞.The choice of c, d > 0 causes the magnitude of this expression to increase without bound as v → ∞.This will later be shown to correspond to h(v) decaying appropriately in this limit.Note that for any specific pdf, such as those considered in Table 1, the choice of H(v) is fixed, and therefore the asymptotic behaviour in the large-v limit can be computed directly.
Differentiating both sides with respect to v and rearranging gives Taking another derivative of both sides gives an asymptotic expression for h(v), Recall that that p(v) ∼ av b and q(v) ∼ cv d as v → ∞, with a = 0 and c, d > 0. We retain only the asymptotically dominant terms in this limit, giving Recall that q(v) ∼ cv d with c, d > 0. It is therefore clear that the asymptotic form for the integral described in (5) corresponds to distributions with exponentially decaying upper tails as v → ∞, corresponding to the small-δ limit.We may therefore write as δ → 0, Using this asymptotic equivalence, (4) can be rewritten as Inverting this equation gives Taking the leading order in the limit that δ → 0 and returning to the original discrete coordinates gives From the definition of a cumulative distribution function, it follows that h(x) decreases monotonically to zero in the limit that x → ∞.Consequently the inverse function h −1 (1/n) grows without bound as n → ∞.This confirms that v n → ∞ as n → ∞, justifying the assumption made at the beginning of this asymptotic analysis.

Asymptotics of v n with Algebraically Decaying Tails
If we return to (5) and set c = 0, we instead pose a polynomial ansatz, and can determine the behaviour of v n for densities, and hence distributions, with algebraically-decaying tails.We instead let c = 0 and p(v) ∼ av b with b > 1.The equivalent expression for (6) is found by setting q(v) = 0, which gives Consequently, the integral behaviour assumed in (5) with c = 0 corresponds to any distribution that has an algebraically decaying upper tail with power b > 1. Recalling the choice of ansatz, we may therefore write for distributions with algebraically decaying tails.
Applying this expression to (4) and rearranging as in Section 2.1.1 gives the asymptotic behaviour The large-n limit is therefore found by considering the large-y limit of h(y).As h(y) decays algebraically as y → ∞, it must be true that v n grows algebraically as n → ∞, validating our earlier assumption.We note this expression is not independent of the asymptotic form of the distribution tail.Instead, we find that the asymptotic behaviour of v n in the large-n limit depends on the algebraic power of the tail decay, although it does not depend on any other feature of the distribution.

Examples
In Table 1, we compute the asymptotic behaviour of v n in the limit that n → ∞ for a number of common pdfs which have support with no upper bound.The exponential, normal, gamma, and Weibull densities all decay exponentially as y → ∞, and the asymptotic behaviour of v n is therefore given by (7).The Pareto density decays algebraically as y → ∞, and in this case the asymptotic behaviour of v n is given by (7).These densities contain a number of parameters, some of which are required to satisfy particular conditions.The exponential distribution requires β > 0, while the normal distribution permits arbitrary µ, but requires σ > 0. The gamma distribution requires α > 0 and β > 0, and the function γ represents the lower incomplete gamma function, discussed in [1].We also assume that α = 1 for the purposes of the calculation shown here.The α = 1 case requires a different analysis, and corresponds to the exponential distribution.The Weibull distribution requires β > 0 and k > 0. The Pareto distribution requires β > 0, and we prescribe that α > 1 in order to satisfy the requirement that h(y) = o(y −1 ) as y → ∞.
The function W denotes the Lambert-W function.The asymptotic expressions for v n associated with the normal and gamma distributions can be further simplified by observing that the argument of the Lambert-W function becomes large in the asymptotic limit, and Using the first two terms of this expansion produces an asymptotic expression with error that is o(1) as n → ∞ for both the normal and gamma distributions.Importantly, for each pdf, we see that v n increases without bound as n → ∞, as assumed and subsequently confirmed by the previous analysis.The asymptotic behaviour of v n will be subsequently used to determine the expectation and variance of τ N for each example.
Table 1: This table details the behaviour of vn in the large-n limit for several common probability density functions.
For each example, the table contains the pdf equation f (y), and the density domain.The next column contains h(y) = 1 − F (y), which shows the rate at which the cumulative distribution function approaches one.The final column shows the asymptotic behaviour of vn in the limit that n → ∞. Gamma: Weibull: 2.2 Calculating optimal stopping statistics

Calculating the expectation
Let w i = P (y < v i ).The expectation is now given by We select a value k such that 0 ≪ k ≪ N ; a representative choice is k = ⌊ √ N ⌋.We write the sum of products as We may use the asymptotic relations in ( 7) and ( 8) to find that where λ = 1 for cdfs with exponentially decaying tails, from (7), and λ = (b − 1)/b for cdfs with algebraically decaying tails having power −b, from (8).While this result holds for a general class of distribution tails h(y), it is straightforward to compute the asymptotic form of w n directly from specific choices of pdf, obtaining identical results.We now consider the first summation term in (10).Recalling that w n = F (v n ) is monotonically increasing in v as n → ∞, and bounded above by one, we see that We can obtain an asymptotic approximation for the second term in the large-N limit, recalling that k is sufficiently large that we can apply the asymptotic expression (10) to each term in the sum.This gives where (a) n = Γ(a + n)/Γ(a), and is known as the Pochhammer symbol [1].By evaluating this symbol directly, we find that Hence, by comparing ( 12) with (11), we see that as N → ∞, Noting this result, the expectation formula (9) gives

Calculating the variance
A similar process may be used to determine the square expectation, and hence the variance.The square expectation is given by We again define k such that 0 ≪ k ≪ N , and split this series to obtain We now consider the first summation term in (14).Recalling that w n = F (v n ) is monotonically increasing in v as n → ∞, and bounded above by one, we see that Using the asymptotic form for w i from (10), noting that k is sufficiently large that this form may be applied, shows that the second term has the asymptotic behaviour This sum can be evaluated to give in the limit that N → ∞.This expression dominates the sum in (15).Hence, the evaluating the asymptotic behaviour of the variance as N → ∞ gives This expression may be simplified to give an asymptotic approximation to the variance,

Examples
In Table 2, we compute the expectation and variance for the common pdfs shown in Table 2, each of which has support with no upper bound.This table illustrates the asymptotics of h(y) in the limit that y → ∞, and the expectation and variance of τ N , obtained using the expectation formula from (13) and the variance formula from (16).
In each of the densities with exponentially decaying tails, the expectation and variance are identical, corresponding to setting λ = 1 in the expectation (13) and variance (16) formulae.This highlights the observation that, despite the four distributions being significantly different, the optimal stopping statistics are entirely determined by the fact that the tail of each probability distribution function decays exponentially in the limit that y → ∞.
We also see that the optimal stopping statistics for the Pareto pdf are different to the preceding examples, as the distribution decays algebraically; consequently, the expectation and variance both depend on the algebraic power of the distribution tail decay rate.

Probability density functions on domains with finite upper bound
We instead define a pdf f (y) on a domain with upper bound y max , having cdf F (y).We again define a function h(y) = 1 − F (y) for subsequent notational convenience.It is clear from the definition of a  contains the asymptotic behaviour of optimal stopping statistics for several common probability density functions, as well as intermediate quantities used to compute these statistics.For each distribution, the table describes the pdf name and domain, the asymptotic behaviour of h(y) in the limit that y → ∞, and the value of λ associated with this asymptotic behaviour.The final two columns contain the expectation and variance of τ N in the limit that N → ∞.
We note from the definition of v n in (1) that, as y is bounded above, the value of v n must also be bounded.We define v max to be the maximum value taken by v n Applying a similar analysis to that which produced (2) gives the recurrence relation In this case, the integral has a finite upper bound, rather than an infinite upper bound.

Large-n asymptotics of v n
We again taking the large-n limit by setting s = n/δ, where s = O(1) and 0 < δ ≪ 1.We define H(y) as the antiderivative of h(y), such that H ′ (y) = h(y).We also define H max = H(y max ).
The governing differential equation is found by applying a similar method to Section 2.1.1,giving where we can no longer discard the term associated with the upper bound, as in (3).This differential equation is separable; by writing H as a function of v, we obtain the solution We now approximate the integral in limit that v → v max , and assume that this corresponds to the small-δ (and hence large-n) limit.As in the infinite domain case, this assumption can be validated upon completing the analysis.
In the limit that v → v max , it must hold that H → H max .It is therefore clear that the integral expression in (17) becomes infinitely large in this limit.We assume it takes the form as v → v max , with b > 0. By differentiating with respect to v and rearranging, we obtain Differentiating again with respect to v gives Hence, the asymptotic form assumed for the integral in (18) corresponds to probability distribution function which approach zero at the upper bound of the domain algebraically.We can relate the integral from (17) to the decay function h in the asymptotic limit, giving Recalling the assumption that v → v max as δ → 0, we can write the differential equation solution (17 Inverting this expression gives Taking the leading order as δ → 0 and returning to discrete coordinates gives Recall that h(y) decreases monotonically to zero in the limit that y → y max .Consequently, as 1/n → 0, the inverse function h −1 (1/n) must tend to v max .This confirms that v n → v max as n → ∞, validating our earlier assumption.The remainder of the procedure is identical to the infinite domain case, with λ = (b + 1)/b.The expectation is therefore given by (13), and the variance is given by (16).We note that these important statistical features depend only on the algebraic rate at which the distribution function approaches the upper bound.

Examples
In Table 3, we compute the asymptotic behaviour of v n in the limit that n → ∞ for a number of common pdfs which have compact support; hence, each of the uniform, triangular, Wigner, and beta distributions are zero above some upper bound.
These pdfs contain a number of parameters.The uniform distribution requires a < b, while the triangular distribution requires a < c < b.The Wigner distribution requires R > 0, and the beta distribution requires α > 0 and β > 0. In the expression for the beta distribution, the function B(α, β) represents the standard beta function, while B(y; α, β) represents the incomplete beta function (see [1]).
We note that the results obtained using this asymptotic formulation for the uniform distribution are consistent with previous analyses from [4,7].
Importantly we see that v n approaches a maximum value for each distribution, corresponding to the maximum possible value of y in the domain, as assumed and subsequently confirmed in the previous analysis.The asymptotic behaviour of v n will be subsequently used to determine the expectation and variance of τ N for each example.
Table 3: This table details the behaviour of vn in the large-n limit for several common probability density functions.
For each example, the table contains the pdf equation f (y), and the density domain.The next column contains h(y) = 1 − F (y), which shows the rate at which the cumulative distribution function approaches one.The final column shows the asymptotic behaviour of vn in the limit that n → ∞.
Table 4: This table contains the asymptotic behaviour of optimal stopping statistics for several common probability density functions, as well as intermediate quantities used to compute these statistics.For each pdf, the table describes the domain, the asymptotic behaviour of h(y) in the limit that y → ymax, and the value of λ associated with this asymptotic behaviour.The final two columns contain the expectation and variance of τ N in the limit that N → ∞.

Conclusions
In this paper we have derived asymptotics of optimal stopping times for sequences of independent identically distributed continuous random variables.In particular, we have found the asymptotics of the expected value and the variance of the stopping time for two large classes of density functions whose domains have either infinite or finite upper bounds.This analysis demonstrated that the asymptotic behaviour of optimal stopping statistics depends on only the behaviour of the upper tail of the cumulative distribution function.Importantly, if a density function f (y) has no upper bound and the upper tail decays exponentially as y → ∞, the expectation and variance of the optimal stopping rule are given by E(τ N ) = N/2 and Var(τ N ) = N 2 /12.These values do not depend on any other features of the distribution, and hold for any distribution with exponentially-decaying upper tail behaviour.
We then considered distributions with no upper bound and algebraically-decaying upper tails.For this class of distributions, we found that the asymptotic behaviour of the optimal stopping statistics depends only on the algebraic power of the distribution upper tail decay rate, and not on any other feature of the distribution.
A similar analysis was applied to distributions on domains with finite upper bounds, showing that the optimal stopping statistics depend only on the distribution behaviour near the upper bound y max .
The key observation produced by this study is that the leading-order asymptotic behaviour of the optimal stopping expectation and variance depends on only the behaviour of the cdf (and hence pdf) near the upper bound.Furthermore, the asymptotic behaviour of these statistics depends solely on the algebraic rate at which the tail approaches the (finite or infinite) upper bound of the distribution domain.
This approach may be extended in straightforward to some discrete density functions; this is illustrated in Appendix A for the geometric density.In general it is more challenging to apply asymptotic methods on discrete domains, and finite discrete domains in particular, requiring intermediate asymptotic matching in order to determine statistical features.The extension of these results to discrete distributions as well as deriving asymptotics of optimal multiple stopping times [10] are a substantial matter for our further research.

A Geometric Distribution
The calculation of expectation and variance for τ N may be performed on distributions with discrete support, although the process is not always so straightforward due to the presence of summation terms rather than integrals in the recurrence relation for v n .This process does still yield convenient solutions for many distributions.For example, if we consider the geometric distribution with parameter p indexed by variable k, we have This gives the recurrence relation We may evaluate this using summation formulae to obtain Using a similar method to the main investigation gives the asymptotic behaviour of v n as Recalling that w n = F (v n ), it is straightforward to show as n → ∞.
and therefore the formulae for expectation and variance hold with λ = p/| log(1 − p)|.
Unfortunately, this asymptotic process becomes substantially more complicated for problems with discrete distributions and a finite upper bound, as v n is bounded and cannot be assumed to be large in the asymptotic limit.Dealing with this limitation requires complicated asymptotic matching techniques.Further work is ongoing on this topic.

Table 2 :
This table