Cramér's Theorem (Large Deviations)¶
An i.i.d.-sample-mean large-deviation principle whose exponential rate is the convex conjugate of the one-observation log moment-generating function.
Core Idea¶
Cramér's theorem gives the large-deviation principle for averages \(\overline X_n\) of i.i.d. real observations when the one-observation log moment-generating function \(\Lambda(t)=\log E[e^{tX_1}]\) is finite near zero. The exponential rate is its convex conjugate \(I(x)=\sup_t\{tx-\Lambda(t)\}\). For a Borel set of possible averages, the lower logarithmic probability bound uses the set's interior and the upper bound uses its closure. A simple limit \(n^{-1}\log P(\overline X_n\ge a)\to-I(a)\) follows for a threshold above the mean when \(I\) is finite and continuous around that threshold; it is not an unconditional formula for every event.[ref-02a40cce71a5][ref-d820259d1eab]
Scope of Application¶
Independent Bernoulli trials give an empirical success fraction with binary-relative-entropy rate. Independent standard Gaussian observations give a sample-average rate \(I(x)=x^2/2\). Both satisfy the theorem's moment condition, but the numerical rates and finite-sample probabilities differ. A fixed-horizon sum of modeled claims or arrivals could qualify if its assumptions are checked; ultimate ruin or all-time overflow is a different event and needs a separate theorem. Cramér's theorem concerns a leading logarithmic rate, not the exact finite-\(n\) tail or its prefactor.[ref-02a40cce71a5][ref-55b77c843ad1]
Clarity¶
State the summand law, verify i.i.d. observations and a log MGF finite near zero, construct its convex conjugate, and identify the event for \(\overline X_n\). Then distinguish the theorem's open/closed-set bounds from a one-line equality: the equality requires the relevant interior and closure rate infima to coincide. A Chernoff calculation alone is only an upper bound; the matching lower bound is the theorem's extra force. Here “exponential rate” means \(n^{-1}\log P\) convergence, not probability divided by \(e^{-nI}\) tending to one.[ref-02a40cce71a5][ref-d820259d1eab][^ref-55b77c843ad1]
Manages Complexity¶
The theorem turns an entire sequence of increasingly rare empirical-mean events into a deterministic rate function derived from a single observation's distribution. The event's least-cost point controls its exponent when the bounds meet. This compression makes unlike distributions comparable, yet deliberately omits finite-sample prefactors and warns that irregular event boundaries can prevent a single limit.[ref-02a40cce71a5][ref-d820259d1eab]
Abstract Reasoning¶
The reusable procedure is: identify i.i.d. real summands; check an exponential moment near zero; form \(\Lambda\) and \(I=\Lambda^*\); apply lower and upper large-deviation bounds to the actual event; only then simplify to a threshold rate if the boundary is regular. For Bernoulli\((p)\) success fractions above \(q\) with \(p<q<1\), this yields \(I(q)=q\log(q/p)+(1-q)\log((1-q)/(1-p))\). For a standard Gaussian average above \(b>0\), it yields \(b^2/2\); a separate Gaussian calculation reveals an additional order-\(n^{-1/2}\) prefactor.[ref-02a40cce71a5][ref-55b77c843ad1]
Knowledge Transfer¶
The construction transfers between discrete and continuous i.i.d. laws because the roles remain the same: observation law, empirical mean, log MGF, convex-conjugate rate and event bounds. It does not transfer automatically to dependent samples, heavy-tailed mechanisms, first-passage events, or a named application merely because they feature rare events. The live Convex Conjugate entry supplies the proposed compositional DAG parent; the theorem adds the domain-specific probability hypotheses and asymptotic conclusion. Neither the broader Large Deviations Theory field nor a one-sided concentration bound duplicates this identity.[ref-02a40cce71a5][ref-d820259d1eab]
[^ref-02a40cce71a5]: Timo Seppäläinen, Translation Invariant Exclusion Processes (2005), Appendix A.7, printed pp. 192–193, Theorem A.9 and Exercise A.9(a). Author-text PDF. [^ref-d820259d1eab]: MIT 6.265/15.070J, Lecture 3: Large Deviations Theory. Cramer's Theorem (2013), printed pp. 1–3. Official course PDF. [^ref-55b77c843ad1]: Jonathan Goodman, Variance Reduction (2005), §4.1, printed pp. 6–7. Author-course PDF.
Relationships to Other Abstractions¶
Current abstraction Cramér's Theorem (Large Deviations) Domain-specific
Parents (1) — more general patterns this builds on
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Cramér's Theorem (Large Deviations) presupposes Convex conjugate Domain-specific
Cramér's rate is defined as the convex conjugate of the one-observation log MGF.
Hierarchy path (1) — routes to 1 parentless root
- Cramér's Theorem (Large Deviations) → Convex conjugate → Duality
Neighborhood in Abstraction Space¶
Cramér's Theorem (Large Deviations) sits in a moderately populated region (50th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Foundations of Probability & Inference (29 abstractions)
Nearest neighbors
- Large deviations theory — 0.87
- Kolmogorov's Three-Series Theorem — 0.87
- Random Variable — 0.86
- Brownian Skorokhod Embedding — 0.85
- Yule–Simon Distribution — 0.85
Computed from structural-signature embeddings · 2026-10-08