# Small Value Probabilities: Techniques and Applications

## Lecture Notes

- Lecture Plan

## Talks

- Oberwolfach Workshop: Small deviation problems for stochastic processes and related topics, Oct. 12 to 18, 2003

# Projects

# Oct. 2012: Small value for Random Sums. $S=\sum_{i=1}^N V_i$

Kalashnikov, V. Geometric Sums: Bounds for Rare Events with Applications. Kluwer
Academic Publishers Group, Dordrecht, 1997.
Barbe, Ph.; McCormick,W. P. Asymptotic expansions for infinite weighted convolutions
of heavy tail distributions and applications. Mem. Amer. Math. Soc. 197 (2009), no. 922,
Nilsen, T.; Paulsen, J. On the distribution of a randomly discounted compound Poisson
process. Stochastic Process. Appl. 61 (1996), 305310.
The large value part has been studied extensively in the literature.
Mentioned to Peng Xu, Oct. 2012
See Tang's N-P (2010) for more.
Watanabe, T. Convolution equivalence and distributions of random sums. Probab. Theory
Related Fields 142 (2008), 367397.
# Sep. 2012: Small value for Products

Let $X$ and $Y$ be two non-negative random variables with distributions $F$ and $G$ on $(0, \infty)$, respectively, and $V=XY$. Find $\P(W \le t)$ as $t \to 0$ based on tail information on $X$ and $Y$.
See refs below for the upper tail case.
Cline, D. B. H. Convolution tails, product tails and domains of attraction. Probab.
Theory Relat. Fields 72 (1986), 529557.
Cline, D. B. H.; Samorodnitsky, G. Subexponentiality of the product of independent
random variables. Stochastic Process. Appl. 49 (1994), 7598.
Foss, S.; Korshunov, D. Lower limits and equivalences for convolution tails. Ann. Probab.
35 (2007), 366383.
Pakes, A. G. Convolution equivalence and infinite divisibility: corrections and corollaries.
J. Appl. Probab. 44 (2007), 295305.
Tang, Q. On convolution equivalence with applications. Bernoulli 12 (2006a), 535549.
Tang, Q. From light tails to heavy tails through multiplier. Extremes 11 (2008), 379
391.
The large value part has been studied extensively in the literature.
See Tang's N-P (2010) for more.
# Aug. 2012: Small value for Perpetuity. $V =^d VM + Q$

#
Charles M. Goldie and Ross A. Maller (2000) Stability of perpetuities. Ann. Probab. 28 1195-1218.
P. Hitczenko On tails of perpetuities, Journal of Applied Probability , 47 (2010), 1191 -- 1194.
P. Hitczenko and G. Letac (2012) Perpetuity property of the Dirichlet distribution,
Maybe to Jennefer Chu, since it is related to Branching.
## References

wli@math.udel.edu